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​ .

$f\left(x\right)=-2x-2$f(x)=−2x−2​

$h\left(x\right)=f\left(5x\right)$h(x)=f(5x)​

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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

The transformation from the graph of f(x)=−2x−2f(x) = -2x - 2 to the graph of h(x)=f(5x)h(x) = f(5x) involves a horizontal compression.

To answer the questions specifically:

  • The graph of hh is a horizontal compression of the graph of ff.
  • The factor of compression is by a factor of 15\frac{1}{5}.

This means that each point on h(x)h(x) is horizontally compressed to be closer to the y-axis by a factor of 5 compared to the corresponding point on f(x)f(x).

Let me know if you need more explanation or have questions about these transformations!

Here are some follow-up questions:

  1. What effect does multiplying xx by a factor greater than 1 have on a graph horizontally?
  2. How would h(x)=f(x2)h(x) = f(\frac{x}{2}) compare in transformation to f(x)f(x)?
  3. What does a negative coefficient in f(x)=−2x−2f(x) = -2x - 2 imply about the graph’s orientation?
  4. How would adding a constant inside h(x)=f(5x)+3h(x) = f(5x) + 3 affect the transformation?
  5. If h(x)=−f(5x)h(x) = -f(5x), what additional transformation occurs?

Tip: Horizontal compressions and stretches depend on factors applied directly to xx, and they are often the inverse of what they appear.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Graph Transformations

Formulas

h(x) = f(ax) results in horizontal compression/stretch based on 'a'

Theorems

Horizontal Compression/Stretches

Suitable Grade Level

Grades 9-12

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