Question

let $f(x) = -2x - 1$. the graph of $f(x)$ is shown below.

a) graph the function $g(x) = f(x) - 4$.

b) graph the function $h(x) = f(x + 2)$.

Explanation:

Step1: Find g(x) equation

Substitute $f(x)=-2x-1$ into $g(x)$:
$g(x) = (-2x-1) - 4 = -2x - 5$

Step2: Identify g(x) key points

• y-intercept: When $x=0$, $g(0)=-2(0)-5=-5$
• x-intercept: When $g(x)=0$, $0=-2x-5 \implies x=-\frac{5}{2}=-2.5$

This is a vertical shift down 4 units from $f(x)$.

Step3: Find h(x) equation

Substitute $x+2$ into $f(x)$:
$h(x) = -2(x+2) - 1 = -2x - 4 - 1 = -2x - 5$

Step4: Identify h(x) key points

• y-intercept: When $x=0$, $h(0)=-2(0)-5=-5$
• x-intercept: When $h(x)=0$, $0=-2x-5 \implies x=-\frac{5}{2}=-2.5$

This is a horizontal shift left 2 units from $f(x)$.

Answer:

a) For $g(x)=f(x)-4=-2x-5$:

• Plot the y-intercept $(0, -5)$ and x-intercept $(-2.5, 0)$, then draw a straight line through them (this is the original $f(x)$ line shifted downward 4 units).

b) For $h(x)=f(x+2)=-2x-5$:

• Plot the y-intercept $(0, -5)$ and x-intercept $(-2.5, 0)$, then draw a straight line through them (this is the original $f(x)$ line shifted left 2 units; note $g(x)$ and $h(x)$ are the same line).