Question

In the xy-plane, the graph of the function g is a parabola. The graph intersects the x-axis at (−4, 0) and (p, 0). If the vertex of g occurs at (−1, −18), what is the value of p ?

Answer 1

Since the vertex of the parabola is given as (-1, -18), the equation of the parabola can be written in vertex form as:

g(x) = a(x + 1)^2 - 18

where "a" is a coefficient that determines the shape of the parabola. Since the parabola intersects the x-axis at (-4, 0) and (p, 0), we know that these two points are the x-intercepts of the parabola.

To find the value of "p", we need to use the fact that the x-intercepts occur when g(x) = 0. Therefore, we can set the equation of the parabola equal to zero and solve for "x":

a(x + 1)^2 - 18 = 0

Simplifying this equation gives:

a(x + 1)^2 = 18

Since the parabola also intersects the x-axis at the point (-4, 0), we know that when x = -4, g(x) = 0. Substituting this point into the equation of the parabola gives:

a(-4 + 1)^2 - 18 = 0

a(3)^2 = 18

9a = 18

a = 2

Now we can substitute this value of "a" into the equation for the x-intercepts:

2(x + 1)^2 - 18 = 0

x + 1 = ±3

Solving for "x" gives:

x = -1 ± 3

x = -4 or x = 2

Since we know that one of the x-intercepts is at (-4, 0), the other intercept must be at (2, 0). Therefore, the value of "p" is 2.

Answer 2

Vertex is (-1,-18), so the axis of symmetry is x = -1.
(-4,0) is 3 units to the left of the axis of symmetry, so (p,0) is 3 units to the right of the axis.
p = -1+3 =2