Math Problem Statement

Use the graphs of $f$f​ and $g$g​ . Which describes the transformation from the graph of $f$f​ to the graph of $g$g​ ?

$f\left(x\right)=-4x+8$f(x)=−4x+8​

$g\left(x\right)=-f\left(x\right)$g(x)=−f(x)​

Responses

horizontal translation 1 units left

horizontal translation 1 units left

vertical translation 4 units down

vertical translation 4 units down

reflection in the x-axis

reflection in the x -axis

reflection in the y-axis

reflection in the y -axis

horizontal shrink by a factor of  $\frac{1}{4}$14​​

horizontal shrink by a factor of 1 fourth

horizontal stretch by a factor of 4

horizontal stretch b y a factor of 4

vertical shrink by a factor of  $\frac{1}{4}$14​​

vertical shrink by a factor of 1 fourth

vertical stretch by a factor of 4

Solution

To analyze the transformation from the graph of f(x)=−4x+8f(x) = -4x + 8 to g(x)=−f(x)g(x) = -f(x), let's go through the transformation step-by-step.

  1. Original Function: f(x)=−4x+8f(x) = -4x + 8

  2. Transformation: g(x)=−f(x)g(x) = -f(x)

    • Substituting f(x)f(x) into g(x)g(x), we get: g(x)=−(−4x+8)=4x−8g(x) = -(-4x + 8) = 4x - 8

Now we have:

  • f(x)=−4x+8f(x) = -4x + 8
  • g(x)=4x−8g(x) = 4x - 8

Identifying the Transformation

The transformation from f(x)f(x) to g(x)g(x) can be described as follows:

  • Reflection in the x-axis: The function g(x)=−f(x)g(x) = -f(x) is obtained by reflecting f(x)f(x) in the x-axis. This means that every point on f(x)f(x) has its y-coordinate negated to get the corresponding point on g(x)g(x).

Therefore, the correct answer is:

  • Reflection in the x-axis

Would you like more details on this transformation or have any other questions?

Here are five related questions:

  1. How does reflecting a function across the y-axis differ from reflecting across the x-axis?
  2. What would be the transformation if g(x)=f(x)+4g(x) = f(x) + 4?
  3. How do horizontal and vertical stretches impact the slope of a linear function?
  4. How does the graph change if f(x)f(x) is multiplied by a negative constant?
  5. Can we describe transformations algebraically for other types of functions, like quadratics?

Tip: For transformations involving reflections, negative signs indicate a flip across an axis: a negative outside the function reflects over the x-axis, and a negative inside reflects over the y-axis.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Reflections
Transformations

Formulas

f(x) = -4x + 8
g(x) = -f(x)

Theorems

Reflection across the x-axis

Suitable Grade Level

Grades 8-10

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