After you find $f(21)$, $f(23)$, and $f(25)$, compare them. For a quadratic function, if two $x$-values are the same distance from some center and their $y$-values match, what does that say about the axis of symmetry and the vertex?

Once you know the vertex $(h,k)$ of $f(x)$, write $f(x)$ in the form $a(x-h)^2 + k$ and plug in one known point to solve for $a$. Then use this formula to evaluate $f(0)$ for the $y$-intercept.

The $y$-intercept happens where the graph crosses the $y$-axis. Which $x$-value does that correspond to, and how can you use your formula for $f(x)$ to find the corresponding $y$-value?

Create a table in Desmos. Enter $21$, $23$, and $25$ in the first column and $-8$, $8$, and $-8$ in the second column. Desmos labels these columns $x_1$ and $y_1$.

On a new expression line, enter y_1~ax_1^2+bx_1+c. This fits the quadratic $ax^2+bx+c$ to the three table points, which represent $f(x)+4$.

Because the fitted quadratic represents $f(x)+4$, define $f(x)$ by entering f(x)=ax^2+bx+c-4 on a new expression line.

Enter f(0). The displayed value is the $y$-coordinate of the $y$-intercept of $y=f(x)$.

The table is for the function $y = f(x) + 4$, but the question asks about the graph of $y = f(x)$.

Use the relationship $y = f(x) + 4$ to find $f(x)$ at the three $x$-values by subtracting 4 from each $y$ in the table:

• When $x = 21$: $f(21) + 4 = -8 \Rightarrow f(21) = -8 - 4 = -12$.
• When $x = 23$: $f(23) + 4 = 8 \Rightarrow f(23) = 8 - 4 = 4$.
• When $x = 25$: $f(25) + 4 = -8 \Rightarrow f(25) = -8 - 4 = -12$.

So the graph of $y = f(x)$ passes through the points $(21,-12)$, $(23,4)$, and $(25,-12)$.

A quadratic function’s graph is a parabola, which is symmetric about a vertical line called its axis of symmetry.

Notice that $(21,-12)$ and $(25,-12)$ have the same $y$-value and are equally spaced around $x = 23$. Therefore, the axis of symmetry is $x = 23$, and the point $(23,4)$ is the vertex of $f(x)$.

The vertex of $f(x)$ is $(23,4)$, and the parabola opens downward because the vertex has the greatest $y$-value among these three points.

The vertex form of a quadratic with vertex $(h,k)$ is

Here, $h = 23$ and $k = 4$, so

Use $(21,-12)$ to solve for $a$:

Thus,

The $y$-intercept occurs when $x = 0$.

So the $y$-coordinate of the $y$-intercept is $-2112$.