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Q. 25

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Precalculus Enhanced with Graphing Utilities
Found in: Page 156
Precalculus Enhanced with Graphing Utilities

Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

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Short Answer

Graph the function f by starting with the graph of y=x2 and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.

Hint: If necessary, write f in the form f(x)=a(x-h)2+k.

f(x)=2x2-4x+1

The required graph is shown below:

See the step by step solution

Step by Step Solution

Step 1. Write the given function in vertex form.

The given function is:

f(x)=2x2-4x+1f(x)=2(x2-2x)+1

Add and subtract the square of half of the coefficient of x inside the parenthesis.

f(x)=2(x2-2x+1-1)+1f(x)=2(x2-2x+1)-2+1f(x)=2(x-1)2-1

Step 2. Determine the transformations used.

In the function f(x)=a(x-h)2+k, a is a constant and (h,k) is the vertex.

In the given function a=2,h=1,k=-1. It means the graph of the given function is a parabola that opens up and has its vertex at (1,-1) and its axis of symmetry is the line x=1.

The graph of y=x2 vertically stretched by factor 2, shifts 1 units right and 1 units down.

First, plot the graph of y=x2 then multiply the values by 2 to get the graph of y=2x2, then shift it 1 unit right to get the graph of the function localid="1646042810491" y=2(x-1)2. After that shift the resulting graph 1 unit down to get the graph of the function f(x)=2(x-1)2-1.

Step 3. Draw the graph.

The required graph is shown below:

Using a graphing utility, we get a graph that is the same as the above graph.

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