Problem 12

Explanation

We analyze each statement based on the graph:

1. Statement $(1)$: The graph intersects the $y$-axis at $4$ and $-2$. So the intersection points are $(0, 4)$ and $(0, -2)$. Statement $(1)$ is CORRECT.

2. Statement $(2)$: Since the graph intersects the $y$-axis at $y = 4$ and $y = -2$, the equation is of the form:

   $$x = a(y - 4)(y + 2)$$

   The graph passes through the point $(8, 0)$ (x-intercept). Substitute into the equation:

   $$8 = a(0 - 4)(0 + 2)$$

   $$8 = a(-4)(2)$$

   $$8 = -8a \implies a = -1$$

   So the function equation is:

   $$x = -(y - 4)(y + 2)$$

   $$x = -(y^2 - 2y - 8)$$

   $$x = -y^2 + 2y + 8$$

   Converting to vertex form:

   $$x = -(y^2 - 2y + 1 - 1 - 8)$$

   $$x = -((y - 1)^2 - 9)$$

   $$x = -(y - 1)^2 + 9$$

   Or in function notation $f(y)$:

   $$f(y) = -(y - 1)^2 + 9$$

   Statement $(2)$ is CORRECT.

3. Statement $(3)$: The axis of symmetry of a horizontal parabola is $y_p$ (ordinate of the vertex). From the form $x = a(y - y_p)^2 + x_p$, we get $y_p = 1$. Or from the midpoint of the roots:

   $$y = \frac{4 + (-2)}{2} = \frac{2}{2} = 1$$

   So the axis of symmetry is $y = 1$. Statement $(3)$ is CORRECT.

4. Statement $(4)$: The intersection with the $x$-axis occurs when $y = 0$. From the graph, it clearly intersects at $x = 8$. By calculation: $x = -(0 - 1)^2 + 9 = -1 + 9 = 8$. So the intersection point is $(8, 0)$. Statement $(4)$ is CORRECT.

Conclusion: All statements $(1)$, $(2)$, $(3)$, and $(4)$ are correct.