SOLUTION: The function h(t)=-16t^2+16t represents the height (in feet) of a horse t seconds after it jumps over a pole.When does the horse reach its maximum height

Question 1206459: The function h(t)=-16t^2+16t represents the height (in feet) of a horse t seconds after it jumps over a pole.When does the horse reach its maximum height
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
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The function h(t)=-16t^2+16t represents the height (in feet) of a horse
t seconds after it jumps over a pole.When does the horse reach its maximum height
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The function is h(t) = -16t^2 + 16t = -16*(t^2-t) = -16*t*(t-1). It is a parabola, having x-intercepts (or t-intercepts) at t= 0 (the horse starts the jump) and t= 1 (the horse completes the jump). The maximum height moment is exactly half way between these time moments at t= 0.5 sec. Another way to find this moment is to use the formula = for the position of the vertex of a parabola y = ax^2 + bx + c. You have then = = = 0.5 sec., the same value.

Solved.

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On finding the maximum/minimum of a quadratic function see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.
Learn the subject from there once and for all.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 0.5 seconds

Explanation
I'll use x in place of t.

The equation we consider is
y = -16x^2+16x

Plug in y = 0 to find the x intercepts.
-16x^2+16x = 0
-16x(x - 1) = 0
-16x = 0 or x-1 = 0
x = 0 or x = 1

The parabola opens downward since the leading coefficient (-16) is negative.
The x intercepts are x = 0 and x = 1.
Due to symmetry, the x coordinate of the max height occurs at the midpoint of those roots.

Add up the x intercept values and divide in half
(a+b)/2 = (0+1)/2 = 1/2 = 0.5
The max height occurs when x = 0.5
This is the final answer since your teacher wants the time value when this max height occurs.

Another way to find this value is to use the formula: h = -b/(2a)
h = x coordinate of vertex
a,b are the first two coefficients of the function.
In this case a = -16 and b = 16.

To find the max height itself, plug x = 0.5 into the function to find:
y = -16x^2+16x
y = -16(0.5)^2+16(0.5)
y = 4
The point (0.5, 4) is the highest point on the parabola.

The horse reaches the max height of 4 feet at the time 0.5 seconds.

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