Question 9

Explanation

Given that the graph of quadratic function $f(x) = ax^2 + bx + c$ intersects the $y$-axis at point $(0, 8)$ and has a vertex at $(2, -4)$.

From point $(0, 8)$, substitute $x = 0$ and $f(0) = 8$ into the function equation

So $c = 8$.

From the vertex $(2, -4)$, the $x$-coordinate of the vertex is $x_p = -\frac{b}{2a} = 2$, so

The $y$-coordinate of the vertex is $f(2) = -4$, so

Substitute $b = -4a$ into the equation $-12 = 4a + 2b$

From $b = -4a$, we get $b = -4(3) = -12$.

So the quadratic function equation is $f(x) = 3x^2 - 12x + 8$.

Statement $\text{(I)}$: The complete equation of the quadratic function is $f(x) = 3x^2 - 12x + 8$

Based on the calculation above, the quadratic function equation is $f(x) = 3x^2 - 12x + 8$. Therefore, statement $\text{(I)}$ is true.

Statement $\text{(II)}$: The point $(3, -1)$ lies on function $f$

Substitute $x = 3$ into the function $f(x) = 3x^2 - 12x + 8$

So $f(3) = -1$, which means the point $(3, -1)$ lies on function $f$. Therefore, statement $\text{(II)}$ is true.

Statement $\text{(III)}$: The value of $f(-1)$ is $22$

Substitute $x = -1$ into the function $f(x) = 3x^2 - 12x + 8$

So $f(-1) = 23$, not $22$. Therefore, statement $\text{(III)}$ is false.

Therefore, the true statements are $\text{(I)}$ and $\text{(II)}$.