Math Problem Statement

What is the formula of g in terms of f? The graph below shows f as a solid blue line and g as a dotted red line. Choose from: A) f(-1/3x), B) f(-3x), C) f(3x), D) f(1/3x).

Solution

To determine the formula of gg in terms of ff, we need to analyze the graph carefully.

Observations:

  1. The graph of ff (solid blue line) is compared to gg (dotted red line).
  2. The red line gg appears compressed horizontally relative to ff. This suggests a horizontal scaling transformation.
  3. Horizontal transformations follow the rule: g(x)=f(kx)g(x) = f(kx) where kk is a scaling factor:
    • k>1k > 1 compresses horizontally.
    • 0<k<10 < k < 1 stretches horizontally.

Analyzing the Compression:

  • On the graph, the solid blue line f(x)f(x) passes through points like (3,0)(3, 0) and (0,6)(0, 6).
  • The dotted red line g(x)g(x) reaches the same y-values as ff, but at scaled-down x-values.

To quantify this scaling:

  • The blue curve ff at x=3x = 3 has the same y-value as the red curve gg at x=1x = 1.
  • This shows the x-values are scaled by a factor of 13\frac{1}{3}.

Thus, the horizontal compression factor k=3k = 3, and the transformation is: g(x)=f(3x).g(x) = f(3x).


Correct Answer:

The formula for gg in terms of ff is C: f(3x)f(3x).


Do you want me to explain horizontal scaling further or have any questions?

Here are 5 related questions for practice:

  1. What happens to f(x)f(x) when it is transformed to f(2x)f(2x)?
  2. How do you determine vertical transformations from a graph?
  3. What is the effect of f(12x)f\left(\frac{1}{2}x\right) on the graph of f(x)f(x)?
  4. What is the difference between vertical and horizontal transformations?
  5. How can you identify a reflection in the xx-axis or yy-axis?

Tip: Horizontal transformations affect the input xx, while vertical transformations affect the output f(x)f(x).

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Horizontal Scaling
Graphs of Functions

Formulas

g(x) = f(kx)
Horizontal compression or stretch factor k

Theorems

Scaling transformations for functions

Suitable Grade Level

Grades 9-12