NO LINKS 1. When a ball was kicked off the ground it landed 20 feet horizontally and reached a maximum height of 6 feet. Create a sketch of the ball's flight and find an equation that describes it.

2. Find the equation of a square foot function with a vertex at (-5, 4) and that passes through the (20, 29).

3. Write the following in graphing form: y = x^2 - 18x + 5

Equation:
Vertex:
x-intercepts:
y-int?

9514 1404 393

Step-by-step explanation:

If we assume the ball was kicked from the origin and that it follows a parabolic curve, the equation can be written ...

  y = -kx(x -20)

for some value of k that makes the maximum be 6. The maximum will occur at the value of x that is halfway between the points where the ball is on the ground, so at x=10. Then our value of k is such that ...

  6 = k(10)(20 -10) = 100k

  k = 6/100 = 0.06

The equation describing the ball's flight is ...

  y = 0.06x(20 -x)

A graph is attached.

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The translated square root function will have a vertical multiplier k that will make it pass through the given point. The parent function f(x) = √x can be translated so its vertex moves from (0, 0) to (-5, 4) by ...

  g(x) = 4 +f(x+5)

Applying the scale factor k gives ...

  g(x) = k·√(x +5) +4

We want g(20) = 29, so ...

  29 = k·√(20 +5) +4

  25 = 5k . . . . subtract 4

  5 = k . . . . . . divide by 5

The equation of the function is ...

  y = 5√(x +5) +4

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We assume your "graphing form" is "vertex form", as that form is generally conducive to graphing.

We can complete the square by adding and subtracting the square of half the x-coefficient:

  y = x^2 -18x +9^2 +5 -9^2

  y = (x -9)^2 -76 . . . . . equation

  vertex: (9, -76)

  x-intercepts: 9±√76 ≈ {0.2822, 17.1778}

  y-intercept: 5 . . . . (the constant in the given equation)

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Additional comment

When the quadratic is written in vertex form ...

  y = a(x -h)^2 +k

the x-intercepts are h±√(-k/a).

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