Question

which function has a range of {y|y ≤ 5}?
○ f(x) = (x − 4)² + 5
○ f(x) = −(x − 4)² + 5
○ f(x) = (x − 5)² + 4
○ f(x) = −(x − 5)² + 4

Explanation:

Step1: Recall vertex form of parabola

Vertex form: $f(x)=a(x-h)^2+k$, where $(h,k)$ is vertex.

Step2: Analyze range based on $a$

If $a>0$, parabola opens up, range $y\geq k$.
If $a<0$, parabola opens down, range $y\leq k$.

Step3: Match required range $\{y|y\leq5\}$

We need $a<0$ and $k=5$.
Check options:

• $f(x)=(x-4)^2+5$: $a=1>0$, range $y\geq5$ (wrong)
• $f(x)=-(x-4)^2+5$: $a=-1<0$, $k=5$, range $y\leq5$ (correct)
• $f(x)=(x-5)^2+4$: $a=1>0$, range $y\geq4$ (wrong)
• $f(x)=-(x-5)^2+4$: $a=-1<0$, $k=4$, range $y\leq4$ (wrong)

Answer:

B. $f(x) = -(x - 4)^2 + 5$