Answer:
[tex]\textsf{A)} \quad (x-40)^2=-100(y-16)[/tex]
B) Focus = (40, -9)
Directrix: y = 41
Axis of symmetry: x = 40
Step-by-step explanation:
The x-intercepts of a parabola are the points at which the curve intercepts the x-axis (when y = 0).
The x-coordinate of the vertex of a parabola is halfway between the x-intercepts.
The y-coordinate of the vertex if the minimum or maximum height of the parabola.
A jumping spider's movement is modeled by a parabola.
Define the variables:
From the information given:
Standard form of a parabola with a vertical axis of symmetry:
[tex](x-h)^2=4p(y-k) \quad \textsf{where}\:p\neq 0[/tex]
If p > 0, the parabola opens upwards, and if p < 0, the parabola opens downwards.
Substitute the vertex (40, 16) and one of the x-intercept points (0, 0) into the formula and solve for p:
[tex]\implies (0-40)^2=4p(0-16)[/tex]
[tex]\implies 1600=-64p[/tex]
[tex]\implies p=-25[/tex]
Substitute the vertex and the found value of p into the formula:
[tex]\implies (x-40)^2=4(-25)(y-16)[/tex]
[tex]\implies (x-40)^2=-100(y-16)[/tex]
Given:
Substitute the given values into the formulas for focus, directrix and axis of symmetry:
Focus
⇒ (h, k+p)
⇒ (40, 16 + (-25)))
⇒ (40, -9)
Directrix
⇒ y = (k-p)
⇒ y = (16 - (-25))
⇒ y = 41
Axis of symmetry
⇒ x = h
⇒ x = 40
