Describe the transformations from f(x)

Question

to g(x)
. f(x)=-1/3x and g(x)=-6 (f(x))-4 Select ALL that apply.

and


A
Shifted down 6

B
Shifted down 4

C
Steeper by a factor of 6

D
Less steep by a factor of 1/3

E
Reflected

Answer 1

To analyze the transformations from \( f(x) = -\frac{1}{3}x \) to \( g(x) = -6(f(x)) - 4 \), we will break down the process step by step.

1. Start with \( f(x) \):

   • \( f(x) = -\frac{1}{3}x \) is a linear function with a negative slope (negative indicates that it's reflected over the x-axis) and a relatively gentle slope because of the factor \( \frac{1}{3} \).

2. Transform \( f(x) \) to \( g(x) \):

   • The transformation starts with \( f(x) \) and is first multiplied by -6:
     \[ -6(f(x)) = -6\left(-\frac{1}{3}x\right) = 2x \] This transformation involves:

     • Steeper by a factor of 6: The steepness of the line changes because multiplying by -6 makes the slope of the line a positive 2 (i.e., the steepness increases, whereas the negative sign indicates it starts going up).

   • Then we shift down by 4:
     \[ g(x) = 2x - 4 \] This is a vertical shift downwards by 4.

Given these transformations, we can summarize the following:

• A: Shifted down 6 (Incorrect. The transformation is a shift down by 4.)
• B: Shifted down 4 (Correct. The function is shifted down by 4 units.)
• C: Steeper by a factor of 6 (Correct. The slope has changed from -1/3 to 2, which is steeper.)
• D: Less steep by a factor of 1/3 (Incorrect. It is not less steep; it is actually steeper.)
• E: Reflected (Correct. The original function is a reflection because of the negative sign in front of \( \frac{1}{3} \) and it maintained a reflection through multiplication.)

Summary of Correct Transformations:

• B. Shifted down 4
• C. Steeper by a factor of 6
• E. Reflected

Therefore, the selected correct options are B, C, and E.