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Quadratic functions

Quadratic functions

In everyday life, one comes across various types of motions of objects, such as the trajectory of a basketball or a volleyball, etc. Such trajectories are in the shape of a ‘parabola’, a curve that can be modeled using a Quadratic Equation of single variable, and such equations of single variables are Quadratic Functions.

Quadratic equations are applied to a variety of practical problems, such as monitoring the path of a projectile, predicting a financial stock model, statistical mechanics, and so on. The part of the equation ax2 is known as the quadratic part, bx as the linear part, and c as the constant part of the function.

Graphing Quadratic Functions - the Parabola

The graph of every Quadratic function is called a parabola.

A parabola is a set of points equidistant from a point and a line.

where the point is called the Focus of a parabola and the line is known as the directrix. Another important point on the parabola is called the Vertex of the parabola. It is the point where the axis of symmetry of a parabola meets the parabola.

Here the axis of symmetry is an imaginary line and the function replicates itself on either side of the line. The graph of a parabola is like a mirror image of a curve, below is a diagram to illustrate this,

Quadratic Functions and Equations graph of a quadratic function StudySmarterThe graph of a quadratic equation - StudySmarter Originals

Here is what the graph of a quadratic function looks like, which is the quadratic function. It can be seen that the curve on the right side of the blue line and the other side of that line, are exactly the same. In mathematical terms, we say that the graph is symmetrical along that blue line. That is why that line is called the axis of symmetry. It is important to note that the axis of symmetry is an imaginary axis, it is not a part of the graph plotted.

Quadratic Functions and Equations graph of a parabola StudySmarterGraph of a Parabola - StudySmarter Originals

It can be seen that the axis of symmetry is parallel to the y-axis and so we say that the parabola is symmetrical to the y-axis. And the point where the parabola meets the axis of symmetry is known as the Vertex of the parabola. It is also the minima of the function. In other words, a vertex is a point where the value of the quadratic function is minimum, hence the name, minima. In the above diagram, point A is the vertex of the parabola.

And for the parabola , the axis of symmetry turns out to be which is symmetrical to y-axis.

There is another crucial point on the parabola, which is the y-intercept of the parabola. It is the point where the parabola meets the y-axis, i.e. where it intercepts the y-axis. Hence, the word, y-intercept. In the above diagram, point C is the y-intercept of the parabola. To find out the coordinates of C, all we need to do is calculate y at x=0. We get,

which gives y=c. Hence, the coordinates of C are (0,c).

Quadratic Function Equations

We can write quadratic function equations in 3 different forms. Let's look at them in more detail

There are three commonly used forms of quadratic functions.

  • Standard or General Form:
  • Factored or Intercept Form:
  • Vertex Form:

Each of these forms can be used to determine different information about the path of a projectile. Understanding the benefits of each form of a quadratic function will be useful for analyzing different situations that come your way.

As the name suggests, the general form is what most quadratic functions are in. The intercept form is useful to easily read off the x and y intercepts of the given curve. The vertex form is especially used when the vertex of the curve has to be read off and determine the related properties.

Standard Form of a Quadratic Function

Quadratic equations in one variable are equations that can be expressed in the form

This is the shape of a parabola, as seen in the image below.

Quadratic Functions and Equations graph of a parabola StudySmarterGraph of a standard parabola - StudySmarter originals

Essentially, these are the equations that have a degree more than Linear equations. Linear equations have a degree of one and quadratic equations have a degree of 2. Here a, b, and c are constants where a≠0. If a=0, then we would only have , which is a linear equation.

So the condition to form a quadratic equation should be that the coefficient of x2 should be non-zero. The other constants b and c can be zero as they won’t affect the degree of the equations.

Vertex Form of a Quadratic Function

The general form of a quadratic may not be the most convenient form to work with, and so we have the Vertex form of a Quadratic Equation. As the name suggests, it is a form based upon the vertex of the parabola formed by the quadratic equation. The vertex is the most important point of a parabola, using which, we can construct the parabola.

The Vertex Form of a Quadratic Equation is given as follows:

where the vertex of the parabola lies at the point (h,k). This form is especially useful when we are given the coordinates of the vertex and are asked to find the equation of the parabola.

Factored Form of a Quadratic Equation

The Factored Form of a Quadratic Equation is a form where the quadratic is factored into its linear factors. Just as we had the vertex form to identify the vertex of a parabola formed by the quadratic equation, the factored form is used to identify the intercepts of the parabola formed.

The Factored or Intercept Form of a Quadratic Equation is given as follows:

where the two x-intercepts are given by . This can be easily verified by setting y=0 and finding the roots of the quadratic equation. Alternatively, one can use the given x-intercepts and a point on the parabola to figure out the quadratic equation.

Examples of Quadratic Functions

Let's practice identifying quadratic functions!

Which of the following are quadratic functions?

(i) (ii) (iii)

Solution:

Recognize the highest degree of each of the functions, if the highest degree is 2 then only it is a quadratic function.

(i)

It can be seen that the highest degree of this function is and it is trivial that and so it is NOT a quadratic function.

(ii)

It is clear that the highest degree of this function is 2 and hence it is a Quadratic function.

(iii)

One can see that the second term has a degree 2 but only the highest degree should be taken into consideration which is 3, and so it is NOT a quadratic function.

Solving Quadratic Equations

Quadratic functions are a generalized form of quadratic equations. When for the quadratic function defined earlier, for some real constant d then the equation formed is known as a Quadratic equation. In general form, a quadratic equation has the form,

where and where represents the set of real numbers. The solution of a quadratic equation is the value of x for which the equation is satisfied. In other words, the solution of a quadratic function is the value of x for which f(x)=0.

We already know that a linear equation has a unique solution, in the case of quadratic equations, there are always two solutions. The solutions need not be unique, they can be the same and solutions may even be complex. However, we will be looking at real solutions and not complex one.

The solutions are also called the zeros of a function. They should not be confused as they are the same thing. To find the zeros, we can simply solve the quadratic using the quadratic formula for zeros, and we get

For practice on how to solve quadratic equations, see our article on Solving quadratic equations and Graph and solve quadratic equations.

The Inverse of a Quadratic Function

Given that a function is Bijective (Injective and Surjective), the inverse exists. For a quadratic function, which is bijective, the inverse of it can be easily calculated. Every inverse is related to the function as follows,

To find the inverse of , we first equate the RHS to y,

The aim is to solve the above quadratic equation in terms of x, i.e., solve for x and express x in terms of y. The above equation can be rearranged to get,

which is quadratic in x, and we can find its roots using the quadratic formula, which gives us,

which is the inverse of y,

Now replacing the variable y with x, we get the inverse in x

where b2+4ax > 4ac for real values of the function.

Quadratic functions - Key takeaways

  • A quadratic function is a function whose highest power is 2. That is the highest degree of the equation is 2.
  • The graph of a quadratic function is called a parabola, with a parent equation of .
  • The solutions (zeros or roots) of a quadratic equation can be calculated using the quadratic formula or factoring the equation into its linear factors.
  • Each quadratic equation has two zeros (they need not be unique). They can be real or imaginary.
  • The graph of a quadratic function is a parabola that can have its axis of symmetry on the y-axis or x-axis.
  • A parabola is defined as the set of points equidistant from a point and a line.
  • One can find the axis of symmetry and the coordinates of the vertex by setting the y=0 and x=0 respectively.

Frequently Asked Questions about Quadratic functions

A quadratic function is a function whose highest power is 2. That is the highest degree of the equation is 2.

Calculate the x-coordinate of the vertex for the quadratic equation y=ax2+bx+c using the formula -b/2a, then substitute this value of x in the original quadratic equation to get the value of y coordinate of the vertex.

The zeros of the quadratic equation y=ax2+bx+c can be found by plugging y=0 in the equation. That is ax2+bx+c=0

Linear and quadratic functions can be solved by plotting graphs. The solution for them would be the point of intersection of both graphs.

The factored form of the quadratic equation is y=a(bx+c)(dx+e)

Final Quadratic functions Quiz

Question

True or False: The equation -3x²-5x-6=0 is a quadratic equation. 

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Answer

True.

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True/False: x²-3x=-5 is a quadratic equation in standard form. 

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Answer

False.

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Question

Given the equation x²-x-6=0, find the y-intercept  

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Answer

-6

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Question

The solution set to the equation x²-4x-5=0 is _____


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Answer

{-1, 5}

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Question

The vertex of the parabola from the function f(x)=x²-4x+5 is _____


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Answer

(2, 1)

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Question

A function that can be written in the form f(x)=ax²+bx+c for real numbers a,b and c, with a≠0 is a _____


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Answer

Quadratic function

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Question

If the discriminant of an equation is 49, what type of solution will it have? 


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Answer

Two rational solutions

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Question

What is the discriminant for the equation 4x²+x+1=0? 


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Answer

-15

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Question

(x-1)(x+1)=0 is known as _____

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Answer

The zero-factor property

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Question

Which equation has the roots (3,9)? 


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Answer

 x²-12x+27=0 

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A quadratic formula is  ______


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a general formula for solving any quadratic equation.  

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If a quadratic equation has only one rational solution what effect does this have on its discriminant? 


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Answer

The discriminant  is 0. 

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The solution to a quadratic equation are synonymous to the ____


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Answer

roots, zeros, or the x-intercepts. 

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Question

What is the factor of the perfect square binomial x²-2x+1? 


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Answer

(x-1)² 

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Question

If the highest degree of an equation is 3, is it a quadratic function?

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Answer

No.

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Question

Is x4-3x+2=y a quadratic function?

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Answer

No.

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Question

What kind of curve is obtained by plotting a quadratic function?

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Answer

Parabola

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What is a Parabola?

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Answer

A parabola is a set of points equidistant from a line and point.

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How many zeros does a quadratic function has?

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Answer

Two

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What are the conditions for a quadratic function (or any function in general) to have an inverse?

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Answer

It should be Injective and Surjective.

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Question

What is the purpose of completing the square of a quadratic equation?

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Answer

To simplify a quadratic equation and determine the maximum or minimum values

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What are the five steps to solving a quadratic equation by completing the squares?

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Answer

  1. Divide the expression by the coefficient of x2
  2. Move the third term to the right-hand side
  3. Complete the square and balance the equation 
  4. Take the square roots of both sides
  5. Solve for x

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Question

If the coefficient of xis positive in a given quadratic equation, what kind of turning point do we obtain?


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Answer

Minimum value

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If the coefficient of xis negative in a given quadratic equation, what kind of turning point do we obtain?

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Answer

Maximum value

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Question

What is the maximum value? What are other terms that describe this?

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Answer

The maximum value is the highest point of the curve in a graph. This is also known as the maximum turning point or local maxima.


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What is the minimum value? What are other terms that describe this?


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The minimum value is the lowest point of the curve in a graph. This is also known as the minimum turning point or local minima.

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What is a vertex? Is there another word that can be used to describe this?

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Answer

The vertex is a point at which the curve on a graph turns. This is also known as a turning point.

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Question

There are three forms for quadratic equations. What form is: y=ax² + bx +c? 

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Answer

Standard form

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Question

The U shape of the graph of a quadratic function is called a...


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Answer

Parabola

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Question

For a quadratic equation with a positive term in x2, what do the coordinates (-ad,e) tell us about the graph?

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Answer

The curve has a minimum value of e at x=-ad

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For a quadratic equation with a negative term in x2, what do the coordinates (ad,e) tell us about the graph?

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Answer

The curve has a maximum value of e at x=ad

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Question

What is the lowest or highest point of a parabola called? 


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Answer

Vertex

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The highest point of a parabola is...... 


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Answer

The maximum

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The lowest point of a parabola is...... 


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Minimum

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Can any quadratic equation be solved by completing the square?

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Answer

Yes, all quadratic equation be solved by completing the square

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The line that splits the parabola in two is called....... 


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Answer

Axis of symmetry

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The x-intercepts of a parabola have three other names. What are they?


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Answer

Zeros, roots and solutions

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Question

In the quadratic equation y=ax²-bx+c, when "a" is positive means? 


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Answer

Upward parabola

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Question

In the quadratic equation y=ax2+bx-c when "a" is negative means? 


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Answer

Downward parabola

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Question

All real numbers are a domain of the quadratic function. True or false


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Answer

True

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Question

The equation y=-(x-h)²+k is in .............. form.  


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Answer

Vertex

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Question

Find the vertex of the equation y=(x+5)²-24 


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Answer

(-5,-24) 

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Question

Convert the equation y=x²+6x+2 into vertex form.


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Answer

(x+3)²=7 

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Question

Find the x-intercepts of the factored form of the equation y=(x+1)(x-5) 


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Answer

(-1,0), (5,0)

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Question

From the factored quadratic equation form y=-(x-3)(x+5), find the values that represent a, b and c in standard form.

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Answer

a=-1, b=-2 and c=15 

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Question

What is the Quadratic Formula used for?

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Answer

To find solutions of a given quadratic equation

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Question

How many solutions does the Quadratic Formula produce? What is the sign in the Quadratic Formula that gives that particular number of solutions?

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Answer

Two solutions. The '±' sign indicates that there are two solutions when we apply the Quadratic Formula.

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Question

How can the Quadratic Formula help us plot the graph of a given quadratic equation?

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Answer

Since the Quadratic Formula determines the roots of a quadratic equation, we can locate them and plot the graph more accurately

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Question

When can we use the Quadratic Formula to solve a given quadratic equation?

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Answer

We can use this for quadratic equations that cannot be factored (however, we can indeed use it to solve any quadratic equation) 

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Question

What is another term used to describe a graph with one real root?

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Answer

Repeated real root

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