Math Problem Statement

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

$f\left(x\right)=\frac{2}{3}x+1$f(x)=23​x+1​

$r\left(x\right)=3f\left(x\right)$r(x)=3f(x)​

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  $r$r​  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)=23x+1f(x) = \frac{2}{3}x + 1 to r(x)=3f(x)r(x) = 3f(x) is a vertical stretch by a factor of 3.

So, for the responses:

  1. The graph of r(x)r(x) is a vertical stretch of the graph of f(x)f(x) by a factor of 3.

Would you like a deeper explanation or have any questions about this? Here are some related questions:

  1. How does a vertical stretch affect the slope and intercept of a linear function?
  2. What would happen if r(x)r(x) were a vertical compression instead?
  3. How would a horizontal stretch or compression be represented in function notation?
  4. What if the transformation was a translation? How would that change the graph?
  5. Could we have any transformations that affect both axes at once?

Tip: When dealing with transformations, note whether each change affects just the output (vertical transformations) or the input (horizontal transformations).

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Transformations
Vertical Stretch

Formulas

f(x) = (2/3)x + 1
r(x) = 3f(x)

Theorems

-

Suitable Grade Level

Grades 8-10

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