9514 1404 393
Answer:
- y = 0.06x(20 -x)
- y = 5√(x+5) +4
- y = (x -9)^2 -76; (9, -76); 9±√76; 5
Step-by-step explanation:
<h3>1.</h3>
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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<h3>2.</h3>
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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<h3>3.</h3>
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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<em>Additional comment</em>
When the quadratic is written in vertex form ...
y = a(x -h)^2 +k
the x-intercepts are h±√(-k/a).
