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.

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The graph of  $h$  is a Response area of the graph of $f$  by a factor of Response area.
$\frac{1}{6}$


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

### Explanation:
The graph of $ f(x) $ is given as $ f(x) = -2x - 6 $, while $ h(x) = \frac{1}{3}f(x) = \frac{1}{3}(-2x - 6) = -\frac{2}{3}x - 2 $.

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

### Final Answer:
The graph of $ h $ is a **vertical compression** of the graph of $ f $ by a factor of **$ \frac{1}{3} $**.

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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."

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(x) equals -2x - 6 goes through the ordered pair (-3, 0) and the ordered pair (0, -6). The line h(x) equals one-third f(x) goes through the ordered pair (-3, 0) and the ordered pair (0, -2).
The graph of h is a transformation of the graph of f by a factor of ____.

Explore the transformation from the graph of f(x) = -2x - 6 to h(x) = (1/3)f(x). Understand how vertical compression affects linear equations and learn the mathematical concepts behind transformations, slopes, and intercepts. Ideal for students in Grades 8-10.

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