Question

which is a feature of function g if $g(x) = -f(x - 4)$?

a. $y$-intercept at $(0, -4)$

b. vertical asymptote of $x = -4$

c. horizontal asymptote of $y = 4$

d. $x$-intercept at $(5, 0)$

Explanation:

Step1: Analyze parent function $f(x)$

From the graph:

• $f(x)$ has vertical asymptote $x=0$
• $f(x)$ has x-intercept at $(1, 0)$
• Horizontal asymptote $y=0$

Step2: Transform to $g(x)=-f(x-4)$

Transformations applied:

1. Horizontal shift right 4 units: replace $x$ with $x-4$
2. Reflection over x-axis: multiply by $-1$

Step3: Find features of $g(x)$

• Vertical asymptote: Shift $x=0$ right 4: $x=4$
• X-intercept: Shift $(1,0)$ right 4 to $(5,0)$, reflect over x-axis: $(5,0)$ (unchanged)
• Horizontal asymptote: Reflect $y=0$ over x-axis: $y=0$
• Y-intercept: Calculate $g(0)=-f(0-4)=-f(-4)$. From $f(x)$ shape (log-like), $f(-4)$ is not $4$, so $g(0)

eq-4$

Step4: Match with options

Only option D matches the derived x-intercept.

Answer:

D. x-intercept at $(5, 0)$