Question

1. displayed below are three distinct parabolas, each plotted with their x-intercepts and vertices clearly marked:

a) observe the position of the vertex in each parabola compared to the position of the x-intercepts. for instance, for the right (purple) parabola, the x-intercepts are at 10 and 16, and the vertex is at x = 13. what do you notice?
the factored form of the equation of the middle (red) parabola is y = (x - 2)(x + 6). how might we use this equation to find the coordinates of the vertex? in other words, if we were only given this equation, how could we figure out the vertex without actually graphing it?

Explanation:

Part (a)

For a parabola, the vertex lies on the axis of symmetry, which is the vertical line that passes through the midpoint of the two \( x \)-intercepts (roots). For the right parabola, the \( x \)-intercepts are 10 and 16. The midpoint of 10 and 16 is \( \frac{10 + 16}{2}=\frac{26}{2} = 13 \), which is the \( x \)-coordinate of the vertex. This pattern holds for other parabolas too: the \( x \)-coordinate of the vertex is the average of the \( x \)-intercepts (roots) of the parabola.

Step 1: Find the \( x \)-intercepts (roots)

For the factored form \( y=(x - 2)(x + 6) \), the roots ( \( x \)-intercepts) are found by setting \( y = 0 \):
\( 0=(x - 2)(x + 6) \)
This gives \( x-2 = 0 \) or \( x + 6=0 \), so the roots are \( x = 2 \) and \( x=-6 \).

Step 2: Find the \( x \)-coordinate of the vertex

As observed in part (a), the \( x \)-coordinate of the vertex is the midpoint of the roots. So, we calculate the midpoint of 2 and -6:
\( x=\frac{2+(-6)}{2}=\frac{2 - 6}{2}=\frac{-4}{2}=-2 \)

Step 3: Find the \( y \)-coordinate of the vertex

Substitute \( x=-2 \) into the equation \( y=(x - 2)(x + 6) \):
\( y=(-2 - 2)(-2 + 6)=(-4)(4)=-16 \)

Answer:

The \( x \)-coordinate of the vertex of a parabola is the midpoint (average) of the \( x \)-intercepts (roots) of the parabola. For example, for the right parabola with \( x \)-intercepts 10 and 16, the vertex's \( x \)-coordinate is \( \frac{10 + 16}{2}=13 \), which matches the given vertex's \( x \)-coordinate.

Part (b)