Write $f(x)$ in the form $a(x-1)(x-5)$. How can you use the y-intercept $(0,20)$ to find the value of $a$?

Plug $x=0$ and $f(0)=20$ into $a(x-1)(x-5)$ and solve for $a$. Once you have $a$, substitute $x=3$ into your function.

In Desmos, type y = a(x - 1)(x - 5). Desmos will automatically create a slider for the parameter a.

Add the point (0,20) as a separate expression. Adjust the a slider until the graph of y = a(x - 1)(x - 5) passes exactly through the point (0,20); this gives you the correct function.

Once the correct value of a is set, either:

• Add a table for y = a(x - 1)(x - 5) and include $x=3$, or
• Tap/click on the graph at $x=3$. Read the corresponding $y$-value; that is the value of $f(3)$.

If a quadratic has x-intercepts at $(r_1,0)$ and $(r_2,0)$, then it can be written as

for some constant $a$.

Here, the x-intercepts are $(1,0)$ and $(5,0)$, so

for some number $a$.

The y-intercept is the point where the graph crosses the y-axis, which happens at $x=0$. We are told the y-intercept is $(0,20)$, so $f(0)=20$.

Substitute $x=0$ into $f(x)=a(x-1)(x-5)$:

Now simplify the right side to solve for $a$.

Continue simplifying:

Divide both sides by $5$:

So the quadratic function is

Now substitute $x=3$ into $f(x)=4(x-1)(x-5)$:

Simplify step by step:

so

Therefore, $f(3) = -16$, which corresponds to choice B.