Question

given: $f(x) = x^2, g(x) = x^2 + 5$
which statement is true about the graphs of $f(x)$ and $g(x)$?
answer
the graph of $f(x)$ is shifted 5 units to the right of the graph of $g(x)$
the graph of $f(x)$ is shifted 5 units up from the graph of $g(x)$
the graph of $g(x)$ is shifted 5 units to the right of the graph of $f(x)$.
the graph of $g(x)$ is shifted 5 units up from the graph of $f(x)$

Explanation:

Step1: Recall Vertical Shift Rule

For a function \( y = f(x) + k \), if \( k>0 \), the graph of \( f(x) \) is shifted \( k \) units up.
Here, \( g(x)=f(x)+5 \) (since \( f(x)=x^2 \) and \( g(x)=x^2 + 5 \)).

Step2: Analyze Each Option

• Option 1: Shifting right involves \( f(x - h) \), not vertical shift. So incorrect.
• Option 2: Shifting right involves \( f(x - h) \), not vertical shift. So incorrect.
• Option 3: \( g(x)=f(x)+5 \), so \( f(x)=g(x)-5 \), meaning \( f(x) \) is shifted down from \( g(x) \), not up. Incorrect.
• Option 4: Since \( g(x)=f(x)+5 \), by vertical shift rule, \( g(x) \) is \( f(x) \) shifted 5 units up. Correct.

Answer:

The graph of \( g(x) \) is shifted 5 units up from the graph of \( f(x) \)