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 parabola range rules

For a quadratic function in vertex form $f(x)=a(x-h)^2+k$:

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

Step2: Match required range condition

We need range $\{y|y\leq 5\}$, so $a<0$ and $k=5$.

Step3: Identify correct function

Check options: $f(x)=-(x-4)^2+5$ has $a=-1<0$ and $k=5$, which fits.

Answer:

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