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Manipulating Functions

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

When we think of manipulating a function, what comes to mind? All those years of math we've taken up until now? Maybe some cobwebs in our brains start to get dusted off and some gears start to turn as we try to remember all the rules, identities, and properties of all these different types of functions, expressions, and equations (and their graphs)? And maybe we get a headache trying to recall all this information?

Well sweat not! We will review all of those things, and more, as we delve into manipulating functions for AP Calculus.

In this article, we will review all the algebra and trigonometry basics needed to manipulate these functions, equations, and expressions. Then, we will introduce how to use all these tools to manipulate algebraic equations, exponential functions, logarithmic functions, and trigonometric functions. Let's get started!

- Manipulating functions: algebra - a quick review
- Manipulating fractions
- Absolute values
- Manipulating powers
- Manipulating roots
- Manipulating logarithms
- Factoring
- Solving quadratic equations

- Manipulating functions: a refresher on trig functions
- SohCahToa
- Special right triangles
- The unit circle
- Graphing trig functions
- Trig identities and formulas

- Manipulating functions: an introduction
- Algebra of functions: algebraic equations and exponential, logarithmic, and trig functions
- Function transformations
- Combining functions
- Symmetry of functions

- Manipulating functions: key takeaways

While functions are at the heart of calculus, algebra, and therefore algebraic manipulation, is the language of calculus. We can’t do calculus without algebra! So, if your memory is the least bit foggy on the rules for fractions, absolute values, powers, roots, logs, factoring, and solving quadratic equations, this is where we review those.

Fractions are everywhere in calculus. No matter how much we want to, we can’t escape them. Here are some rules for, properties of, and techniques to use with fractions that we need to remember:

Rules:

Never divide by zero! If the denominator of a fraction is zero, then it is undefined.

BUT, the numerator of a fraction can be zero.

Any fraction with a zero in the numerator is equal to zero, as long as zero is not in the denominator.

When adding or subtracting fractions, the denominators must be equal.

To add or subtract fractions, the Least Common Denominator (LCD) must be found.

If the denominator of a fraction is 1, then the fraction simplifies to an integer.

If the numerator and denominator of a fraction are equal, then the fraction simplifies to 1.

Rule of equivalent fractions:

Two fractions and are equal and can be written as if

Remember the reciprocal. If we flip a fraction upside down, we get its reciprocal.

Multiplying a fraction by its reciprocal always gives us 1, provided the numerator of the original fraction is not zero.

Properties:

Commutativity of addition with fractions:

Commutativity of multiplication with fractions:

Associativity of addition with fractions:

Associativity of multiplication with fractions:

Distributivity of multiplication over addition with fractions:

Techniques:

We can use multiplication or division to make equivalent fractions:

We can write integers as fractions:

Adding fractions:

, for fractions with the same denominator

, for fractions with different denominators

Subtracting fractions:

, for fractions with the same denominator

, for fractions with different denominators

Multiplying fractions:

Dividing fractions:

Simplifying (also called canceling) fractions:

It is important to remember that we can only cancel in a fraction when it has an unbroken chain of multiplication throughout the entire numerator and denominator.

can be simplified to , but is in its simplest form (

**cannot**cancel here).

Taking the absolute value of a number means turning a negative number positive, and doing nothing to a positive number or zero. For example:

The straight lines surrounding the number means “the absolute value of”. So when we see this notation, we know to take the absolute value of the number or variable inside.

The rules of powers (also known as exponents) come in very handy in our study of AP Calculus. They make solving problems much easier. And don’t forget, these rules work in reverse, too! But first, let's recap a couple of definitions.

Let's take the number 2 and multiply it by itself 4 times. We get:

This gets tedious to write out, especially if we want to multiply a number by itself many times. So, we use exponents or powers to write this more concisely:

This is read as "two to the fourth power equals sixteen" or "two to the power of four equals sixteen."

**Exponents **or **Powers **are used to show the repeated multiplication of a number (or variable) by itself. In our example above, 4 is the exponent (or power). It tells us how many times to multiply the number 2 times itself.

The **Base** is the number (or variable) that is being multiplied. In our example above, 2 is the base.

Now, let's recap the power rules. For these, we need to make a couple of assumptions:

We assume X and Y are nonzero real numbers.

We assume that m and n are integers.

The power rules are listed as follows:

Product rule:

Quotient rule:

Negative exponent rule:

Zero power rule:

Power of a power rule:

Product to a power rule:

Quotient to a power rule:

Fractional power rule:

**But be careful!** **powers do not distribute in this way**. This type of power must be multiplied out the long way. Does the FOIL method ring a bell?

As you might have noticed from the power rules, powers and roots are related. This relationship is extremely useful when solving and simplifying AP Calculus problems. Basically, any root can be converted to a power, and vice versa. The root rules are summarized below.

The number under an

**even root**(2, 4, 6, etc.)**can’t****be negative**!But the number under an

**odd root**(3, 5, 7, etc.)**can****be negative**.For roots where n is an even number:

For roots where n is an odd number:

Zero root rule:

Identity root rule:

Product rule:

Quotient rule:

Root to a root rule:

Rationalizing the denominator:

By convention, we don't want roots in the denominator of a fraction. We can remove them by a process called rationalization, where we multiply the fraction with a root in its denominator by another fraction (that otherwise would simplify to 1) with that same root in its denominator so that the root is canceled out. For example:

Roots can be rewritten as powers. It is common practice in AP Calculus to first rewrite a problem with roots in it as a problem with powers instead because powers are more visually appealing and a bit easier to understand.

A logarithmic expression is another way of writing an exponential expression. The base of a logarithmic expression can be any positive number, except for 1. If we want to use e as the base of a logarithmic expression, we write ln instead of log. If we want to use 10 as the base of a logarithmic expression, by convention, we don’t write it. The rules for logarithmic expressions are summarized below.

Zero log rule:

Identity log rule:

Product rule:

Quotient rule:

Power rule:

Log of a power rule:

Power of a log rule:

Logarithms and powers (or exponents) are inverses of each other. It is also common practice in AP Calculus to rewrite logarithms as powers, again because powers are more visually appealing and a bit easier to understand.

The formula to convert logs to powers (and vise versa) is:

Another super helpful tool for AP calculus is factoring. Factoring is like breaking a number into its prime factors, but for algebraic expressions. Factoring an expression involves rewriting a sum of terms as a product of terms. We do this by pulling out - or factoring - the Greatest Common Factor (GCF) from all the terms of the expression.

Let's look at the expression:

Each term in the expression has as a factor. So, this GCF can be pulled out of the expression, and it can be rewritten as:

Once we pull out the GCF, the next step is to look for one of the following patterns:

Difference of squares -

**it is absolutely necessary to know how to factor this!**A difference of squares can be factored, but a sum of squares cannot.

Sum of cubes

Difference of cubes

Square of the sum of two numbers

Square of the difference of two numbers

Cube of the sum of two numbers

Cube of the difference of two numbers

Square of the sum of three numbers

Square of the difference of three numbers

Going back to algebra days, a quadratic equation is a second-degree polynomial. Quadratic equations can be solved by one of the following methods.

If it is possible, factoring a quadratic equation is perhaps the easiest and quickest way to solve a quadratic equation. We can test if a quadratic equation is factorable by finding its discriminant. If the discriminant is a perfect square, then the quadratic equation is factorable.

Solve the quadratic equation:

Solution:

- Move all the terms to the left side of the equals sign.

- Factor, using the FOIL (First, Outer, Inner, Last) method to double-check they are correct.

- Set each factor equal to zero and solve for x.

Regardless if a quadratic equation is factorable, we can always solve it using the quadratic formula. For a quadratic equation of the form:

We can find the solutions using the quadratic formula:

Using the same equation as the previous example, solve the quadratic using the quadratic formula.

Solution:

- Move all the terms to the left side of the equals sign.

- Plug the coefficients into the formula and solve.
- In our case, a is 2, b is -5, and c is -12.

This method for solving a quadratic equation is aptly named because it involves taking the quadratic formula in question and making it a perfect square so that it is solvable by taking the square root. We can always use this method to solve a quadratic equation as well, regardless of if it is factorable.

Solve the quadratic equation:

Solution:

- Move all terms with an x in them to the left side of the equals sign, and the constant on the right side.

- Divide both sides of the equation by the coefficient of if it isn't 1.

- Now for the tricky part: take the coefficient of
**x**(**not**) , divide it in half, square it, and then add the result to both sides of the equation.- In our case, the coefficient of x is 8.
- Half of -8 is -4. -4 squared is 16. So, we add 16 to both sides of the equation:

- In our case, the coefficient of x is 8.

- Factor the left side of the equation into a squared binomial.
Note that the factor is always the coefficient of x, divided in half.

- Finally, take the square root of both sides of the equation and simplify.
Remember that the right side needs a plus/minus sign.

Many of the problems we will encounter in AP Calculus involve manipulating trig functions. So, for those of us who don't want to have to relearn trigonometry at the same time as learning AP Calculus, this refresher is key.

Trigonometry starts with triangles - specifically right triangles. The image and table below summarizes key terms of right triangles and the six main trig functions.

The 6 main trig functions | |

The main trig functions are defined by ratios of the sides of a right triangle in a specific order. The side labeled Hypotenuse (or just **H **for short) is always the triangle's longest side. The side labeled Adjacent (or just **A **for short) is the side that is touching the angle of the triangle we are considering (which is θ in the image above). The side labeled Opposite (or just **O** for short) is the side that is across from the angle. **SohCahToa **(pronounced "So-Kah-Toe-Ah") is a mnemonic device used to help us remember the ratios that make up the sine, cosine, and tangent trig functions.

SohCahToa mnemonic device | ||

Soh | Cah | Toa |

From there, we can remember cosecant, secant, and cotangent as the reciprocals of sine, cosine, and tangent.

SohCahToa reciprocals | ||

With that in mind, there are two special right triangles that we deal with in a lot of AP Calculus problems that deserve mention.

Also called an **isosceles right triangle**, the **45-45-90 triangle** is exactly what it sounds like: a right triangle whose other two angles are 45 degrees. What is special about this triangle is that the two legs of the triangle that touch the right angle are always the same size, and the hypotenuse is always times longer than the two legs. It's a good idea to memorize the trig functions for this triangle.

The **30-60-90 right triangle** is also aptly named. What is special about this triangle is that the hypotenuse is always twice as long as the shortest leg, and the longer leg is always times longer than the shorter leg. It's also good to memorize the trig functions for this triangle for both the and angles.

A more convenient way to view the trig functions of these angles (plus angles of 0 and 90 degrees) is with a table.

angle (θ) in degrees | angle (θ) in radians | sin(θ) | cos(θ) | tan(θ) | csc(θ) | sec(θ) | cot(θ) |

0° | 0 | 0 | 1 | 0 | undefined | 1 | undefined |

30° | π/6 | 2 | |||||

45° | π/4 | 1 | 1 | ||||

60° | π/3 | 2 | |||||

90° | π/2 | 1 | 0 | undefined | 1 | undefined | 0 |

While the SohCahToa mneumonic device is great for right triangles, it doesn't work for angles at or greater than 90 degrees. For those, we need the unit circle. The unit circle allows us to find trig values for any angle, not just acute ones. The unit circle has a radius of one unit (hence the name) and is set in the x-y coordinate plane.

The above image tells us quite a lot. So let's walk through it a bit. To start, when we measure angles using the unit circle, we start at the positive x-axis and rotate counterclockwise (see the angle in the image). If we rotated clockwise, we would get a negative angle (see the angle in the image). This is true regardless of whether we are measuring the angle in degrees or radians. In AP Calculus, radians are almost always the preferred way to measure angles. Remember, the radian measure of an angle is the length of the arc along the circumference of the unit circle (see the bolded edge of the circle labeled ). The easiest way to convert between degrees and radians is the formula:

A closer look at the unit circle above reveals the two 30-60-90 right triangles. One is in the first quadrant of the circle, the other is in the second quadrant. Based on these two triangles, we can gather that **the coordinates on the unit circle tell us the angle's cosine and sine values**.

The endpoint of an angle on the unit circle gives us, in order, the angle's cosine and sine values. The x-coordinate is the cosine value, and the y-coordinate is the sine value. We can remember the order by remembering x and y are in alphabetical order just like cosine and sine.

With this in mind, let's see the unit circle with all of the cosine and sine values plotted for the 30-60-90 and 45-45-90 triangles.

It is important to note that the values of all the main trig functions are positive in the first quadrant, only sine and cosecant are positive in the second quadrant, only tangent and cotangent are positive in the third quadrant, and only cosine and secant are positive in the fourth quadrant of the unit circle. Another helpful mnemonic device to remember which functions are positive in what quadrants is **All Students Take Calculus**.

**All**-**All**6 trig functions are positive in the first quadrant.**S**tudents**Sine**(and its reciprocal, cosecant) are positive in the second quadrant.**T**ake**Tangent**(and its reciprocal, cotangent) are positive in the third quadrant.**C**alculus**Cosine**(and its reciprocal, secant) are positive in the fourth quadrant.

The main trig functions - sine, cosine, and tangent - and their reciprocals - cosecant, secant, and cotangent - are periodic functions. This means their graphs are the same shape repeated over and over again, indefinitely. The period of these functions is the length of one of their cycles.

Now that we have refreshed the unit circle, we can sketch the graphs of sine, cosine, and tangent easily by noting some important facts:

For sine and cosine:

The graphs of sine and cosine are the same shape.

The cosine graph is just shifted to the left 90 degrees.

The sine and cosine graphs have a max of 1 and a min of -1.

The sine and cosine graphs repeat their shapes indefinitely to the left and right.

Their shapes repeat every 360 degrees, so their period is 360 degrees (which is the same number of degrees a circle has)

The unit circle tells us that:

Using these 5 points, we can sketch one cycle of the sine and cosine functions.

For tangent:

The basic shape of the graph is a backward S

This shape is repeated indefinitely to the left and right.

It repeats every 180 degrees, so its period is 180 degrees.

Because , we can use the unit circle to sketch the graph:

Because , we draw vertical asymptotes at those points.

The most common way we will want to solve AP Calculus problems that involve trig functions is by using trig identities. Several "gotcha" moments in AP Calculus revolve around us remembering that these identities exist! Plus, trig identities help us solve more complex calculus problems much more quickly and easily. So, let's list them right here.

First, recall that the tangent is a ratio of sine/cosine, and therefore cotangent is the ratio of cosine/sine.

Remember which functions are reciprocals of each other.

The Pythagorean trig identities are probably some of the most useful trig identities. These are based on the Pythagorean Theorem.

Identity 2 is derived by dividing identity 1 by .

Identity 3 is derived by dividing identity 1 by .

These are also known as even/odd identities. As you can see, cosine and secant are the only two even functions, the rest of the trig functions are odd functions.

If n is any integer, then the following identities hold true.

These come from the fact that the period of sine and cosine and their reciprocals is , and the period of tangent and cotangent is , so shifting these by this much left or right results in the same function.

These show which trig functions are complements of each other.

Sine and

**co**sine are**co**mplements, so they are**co**functions of each other.Tangent and

**co**tangent are**co**mplements, so they are**co**functions of each other.Secant and

**co**secant are**co**mplements, so they are**co**functions of each other.

The double angle trig identities are also very useful to remember.

The half-angle trig identities are also very useful to remember. The plus or minus depends on in which quadrant the original given value exists.

The sum and difference of angles trig identities are helpful for simplifying complex-looking expressions.

The sum-to-product trig identities take a sum of two trig functions and convert them to a product of trig functions.

The product-to-sum trig identities take a product of two trig functions and convert them to a sum of trig functions.

For triangles that aren't right triangles, we also have the Law of Sines, the Law of Cosines, and the Law of Tangents.

Law of Sines | Law of Cosines | Law of Tangents | Mollweide's Formula |

Now that we've got all our review out of the way, let's introduce the ways we can manipulate functions. There are four main ways we can manipulate functions:

- Algebra of functions (algebraic manipulation): algebraic equations and exponential, logarithmic, and trig functions
- Think: addition, subtraction, multiplication, and division

- Function transformations
- Think: manipulating a graph

- Combining functions
- Symmetry of functions

The first way we can manipulate functions is by algebraic manipulation. This usually means adding, subtracting, multiplying, and/or dividing a value from a function, or adding, subtracting, multiplying, and/or dividing two or more functions. It can also mean taking a function to a power, root, log, etc. Algebraic manipulation can even mean simplifying an expression, equation, or function. We can algebraically manipulate algebraic equations, exponential functions, logarithmic functions, and trig functions.

Before we get into manipulating algebraic equations, let's review what exactly an algebraic equation is.

An **Algebraic Equation** is like a scale: two expressions, or quantities, sit on either side of an equals sign. The equals sign means that the expression on the left and the expression on the right must be equal to each other.

In other words, "equation" means "equality". Algebraic equations involve equating one quantity with another.

Essentially what we have to do here is make sure we keep both sides of the equals sign equal, as we would with a balance scale. What we do to one side of the equation, we must do to the other side to keep the balance!

Before we get into manipulating exponential functions, let's review what exactly an exponential function is.

An **Exponential Function** is a function that looks very similar to a quadratic function, but with a very important twist: the independent variable and exponent are switched!

If b is any real number greater than 0, **but not 1**, then an exponential function is a function of the form,

where b is called the base and x is any real number.

Again, what we need to do here is make sure we treat the equals sign like a scale, just like we did with the algebraic equations.

Before we get into manipulating logarithmic functions, let's review what exactly a logarithmic function is.

A **Logarithmic Function** is the inverse of the exponential function and therefore has a similar definition.

If b is any real number greater than 0, but not 1, then a logarithmic function is a function of the form,

where b is called the base, and x is any real number.

Again, what we need to do here is make sure we treat the equals sign like a scale, just like we did with the algebraic equations.

Before we get into manipulating trig functions, let's review what exactly a trig function is.

A **Trig Function** (also called a circular function) is a function of a circle (or just an arc) or angle that, at its core, is most simply expressed in terms of the ratios of two sides of a right triangle.

The 6 main trig functions are:

**Sine**and its reciprocal**Cosecant****Cosine**and its reciprocal**Secant****Tangent**and its reciprocal**Cotangent**

Again, what we need to do here is make sure we treat the equals sign like a scale, just like we did with the algebraic equations.

For more information, check out our Algebra of Functions article! There, we cover these topics much more in-depth, and with plenty of examples.

All the function transformations you have learned up until now still apply in AP Calculus. Any function can be transformed, both horizontally and/or vertically, by shifting, reflecting, stretching, or shrinking it. Horizontal transformations only change the x-coordinates of points. Vertical transformations only change the y-coordinates of points.

Horizontal transformations are made when we either add/subtract a number from a function's input variable, usually x, or multiply x by a number. Horizontal transformations, except reflection, work in the **opposite **way we'd expect them to. Here is a summary of how horizontal transformations work:

Shifts - Adding a number to x shifts the function to the left, subtracting shifts it to the right.

Shrinks - Multiplying x by a number greater than 1 shrinks the function horizontally (think horizontal compression).

Stretches - Multiplying x by a number less than 1 stretches the function horizontally.

Reflections - Multiplying x by -1 reflects the function horizontally (over the y-axis).

Vertical transformations are made when we either add/subtract a number from the entire function, or multiply the entire function by a number. Unlike horizontal transformations, vertical transformations work the way we expect them to (yay!). Here is a summary of how vertical transformations work:

Shifts - Adding a number to the entire function shifts it up, subtracting shifts it down.

Shrinks - Multiplying the entire function by a number less than 1 shrinks the function vertically (think vertical compression).

Stretches - Multiplying the entire function by a number greater than 1 stretches the function vertically.

Reflections - Multiplying the entire function by -1 reflects it vertically (over the x-axis).

For more information, check out our Function Transformations article! There, we cover the topic much more in-depth, and with plenty of examples.

Combining functions is another way to manipulate functions in AP Calculus. There are two main ways to combine functions:

Using algebraic manipulation (as described above) to add/subtract/multiply/divide two or more functions.

Substituting the independent variable of one function with another function in a process called function composition.

Since we talked about algebraically manipulating functions already, let's discuss function composition here. Function composition involves taking one function, say , and plugging it into another function, say , and then solving, usually for a value of x.

This also comes with its own notation: both of which are read as "f of g of x".

It is important to note that when combining functions, the order matters!

In other words, .

For more information, check out our Combining Functions article! There, we cover the topic much more in-depth, and with plenty of examples.

Certain functions display properties of **symmetry **that help us understand them and the shapes of their graphs. There are two types of symmetry when we talk about functions and their graphs:

- Even - If a function has
**even**symmetry, that means it is**symmetrical with respect to the y-axis**. - Odd - If a function has
**odd**symmetry, that means it is**symmetrical with respect to the origin**.

For more information, check out our Symmetry of Functions article! There, we cover the topic much more in-depth, and with plenty of examples.

- To be able to manipulate functions, we need to review algebra and trig concepts.
- There are 4 main ways to manipulate functions.
- Algebraic manipulation (algebra of functions):
- The addition, subtraction, multiplication, and division of functions, or
- Raising a function to a power, taking it to a root or absolute value, or
- Any other algebraic operation we can think of.

- Function transformations (manipulating the graphs of functions):
- This happens in two ways:
- Horizontal transforms of a graph.
- Vertical transforms of a graph.

- This happens in two ways:
- Combining functions (function composition):
- Plugging one function into another and solving for a variable.

- Symmetry of functions:
- Knowing how the graph of a function should look based on a function's properties of symmetry.
- There are two types of symmetry when we talk about functions and their graphs:
- Even - the graph is symmetrical with respect to the y-axis.
- Odd - the graph is symmetrical with respect to the origin.

- Algebraic manipulation (algebra of functions):
- For AP Calculus, it is important to know how to:
Manipulate algebraic equations

Manipulate exponential functions

Manipulate logarithmic functions

Manipulate trig functions

There are three main ways to manipulate a function.

- Algebraic manipulation
- Function composition
- Transformations

Manipulating functions using algebra could mean adding, subtracting, multiplying, or dividing a value from a function, or it could mean adding, subtracting, multiplying, or dividing two or more functions together.

Manipulating functions using function composition, or combining functions, means using one function as an input to another and then solving.

Manipulating functions using transformations means algebraically manipulating either the entire function or just the independent variable to change how the graph of the function looks.

You can manipulate rational functions by using algebraic manipulation, function composition, and/or transformations.

But be careful! You must avoid having 0 in the denominator of a rational function!

More about Manipulating Functions

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