Math Problem Statement

Use the graphs of $f$f​  and  $h$h​  to describe the transformation from the graph of $f$f​  to the graph of  $h$h​ .

$f\left(x\right)=-x+1$f(x)=−x+1​

$h\left(x\right)=f\left(2x\right)$h(x)=f(2x)​

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The graph of  $h$h​  is a Response area of the graph of $f$f​  by a factor of Response area.

Solution

To describe the transformation from the graph of f(x)=−x+1f(x) = -x + 1 to h(x)=f(2x)h(x) = f(2x):

  1. The graph of h(x)h(x) is a horizontal compression of the graph of f(x)f(x).
  2. The compression factor is by a factor of 12\frac{1}{2}.

So, the complete statement would be:

  • The graph of h(x)h(x) is a horizontal compression of the graph of f(x)f(x) by a factor of 12\frac{1}{2}.

Would you like a detailed explanation of this transformation?

Here are five related questions:

  1. What happens to a function when it is vertically stretched or compressed?
  2. How would the transformation change if h(x)=f(x2)h(x) = f(\frac{x}{2})?
  3. What effect does a negative sign in h(x)=−f(x)h(x) = -f(x) have on the graph?
  4. How do you determine the horizontal shift of a function?
  5. What is the impact on the graph if h(x)=f(x)+ch(x) = f(x) + c where cc is a constant?

Tip: For horizontal transformations, multiplying xx by a factor greater than 1 compresses the graph, while dividing xx stretches it.

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Graphing Functions
Horizontal Compression

Formulas

h(x) = f(kx) results in a horizontal compression if |k| > 1

Theorems

Transformation Theorem for functions

Suitable Grade Level

Grades 9-11

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