Question

write a function g whose graph represents the indicated transformation of the graph of f.

1. $f(x) = 4x + 1$; translation 2 units left
2. $f(x) = -4|x - 2|$; vertical shrink by a factor of $\frac{1}{2}$
3. let g be a translation 4 units down and a horizontal shrink by a factor of $\frac{1}{4}$ of the graph of $f(x) = x$.
4. let g be a reflection in the x - axis and a vertical stretch by a factor of 3, followed by a translation 4 units down 1 unit right of the graph of $f(x) = |x|$.

Explanation:

Problem 8

Step1: Recall horizontal translation rule

For a function \( y = f(x) \), a translation \( h \) units left is given by \( y = f(x + h) \). Here, \( h = 2 \) and \( f(x)=4x + 1 \).

Step2: Apply the translation

Substitute \( x + 2 \) into \( f(x) \): \( g(x)=f(x + 2)=4(x + 2)+1 \).
Simplify: \( g(x)=4x+8 + 1=4x + 9 \).

Step1: Recall vertical shrink rule

For a function \( y = f(x) \), a vertical shrink by a factor of \( k \) (where \( 0

Step2: Apply the vertical shrink

Multiply \( f(x) \) by \( \frac{1}{2} \): \( g(x)=\frac{1}{2}\cdot f(x)=\frac{1}{2}\cdot(-4|x - 2|) \).
Simplify: \( g(x)=-2|x - 2| \).

Step1: Recall horizontal shrink and vertical translation rules

• Horizontal shrink by a factor of \( \frac{1}{a} \) (here \( a = 4 \)) for \( y = f(x) \) is \( y = f(ax) \).
• Vertical translation \( k \) units down (here \( k = 4 \)) is \( y = f(ax)-k \).

Step2: Apply the transformations

First, horizontal shrink: \( f(4x)=4x \) (since \( f(x)=x \)).
Then, vertical translation down 4 units: \( g(x)=4x-4 \).

Answer:

\( g(x)=4x + 9 \)

Problem 9