Math Problem Statement

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

Two linear equations graphed on a coordinate plane. The line f of x equals negative 2 x minus 6 goes through ordered pair negative 3 comma 0 and ordered pair 0 comma negative 6. The line h of x equals one-third f of x goes through ordered pair negative 3 comma 0 and ordered pair 0 comma negative 2.

Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. The graph of $h$ is a Response area of the graph of $f$ by a factor of Response area. $\frac{1}{6}$

Solution

The transformation from the graph of f(x)=−2x−6f(x) = -2x - 6 to the graph of h(x)=13f(x)h(x) = \frac{1}{3}f(x) involves a vertical compression.

Explanation:

The graph of f(x)f(x) is given as f(x)=−2x−6f(x) = -2x - 6, while h(x)=13f(x)=13(−2x−6)=−23x−2h(x) = \frac{1}{3}f(x) = \frac{1}{3}(-2x - 6) = -\frac{2}{3}x - 2.

Comparing these equations:

  • The slope of f(x)f(x) is −2-2, and the slope of h(x)h(x) is −23-\frac{2}{3}, which is smaller in magnitude, indicating a vertical compression of the graph.
  • The y-intercepts are −6-6 for f(x)f(x) and −2-2 for h(x)h(x), further confirming this compression.

Final Answer:

The graph of hh is a vertical compression of the graph of ff by a factor of 13\frac{1}{3}.

Do you want more details or have any questions about this transformation?

Here are 5 questions to deepen your understanding:

  1. How do we determine if a transformation is a vertical stretch or compression?
  2. What effect does multiplying a function by a factor less than 1 have on its graph?
  3. How does changing the slope affect the steepness of a line on a graph?
  4. How would the transformation change if the factor were negative?
  5. What happens to the y-intercept during a vertical compression?

Tip: Vertical compression occurs when the factor is between 0 and 1, making the graph appear "flatter."

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Transformations
Slope-Intercept Form

Formulas

f(x) = -2x - 6
h(x) = (1/3)f(x) = -(2/3)x - 2

Theorems

Vertical Compression

Suitable Grade Level

Grades 8-10

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