下图显示了函数图像 y = -x^2 + p x + q 的草图。 该图像与 x 轴交于点 A 和 B(3, 0)。 该图像还与 y 轴交于点 C。 该图像对称轴的方程为 x = -1。

1. Use the axis of symmetry to find the other x-intercept.

Since the line of symmetry is $x=-1$, it is midway between the x-coordinates of the two roots.

Let $A=(a,0)$. Given $B=(3,0)$,
$\frac{a+3}{2}=-1$
$a+3=-2$
$a=-5$

Hence $A=(-5,0)$.

1. Write the quadratic in factorised form using its x-intercepts.

From part (a), the x-intercepts are $x=-5$ and $x=3$.

Since the coefficient of $x^2$ in $y=-x^2+px+q$ is $-1$,
$y=-(x+5)(x-3).$

2. Expand the factorised expression.

$y=-(x+5)(x-3)$
$y=-(x^2+2x-15)$
$y=-x^2-2x+15.$

3. Compare coefficients with the given form to find $p$ and $q$.

Compare
$y=-x^2-2x+15$
with
$y=-x^2+px+q.$

Hence,
$p=-2,\qquad q=15.$

1. Use the y-intercept of the quadratic to write the coordinates of $C$.

At the y-axis, $x=0$.

From part (b), $q=15$. Since the graph is $y=-x^2+px+q$, the y-intercept is

$C=(0,q)=(0,15).$