The parabola y = -16x^2 + 32x opens downward and the y value of the vertex is 16
The direction of the parabola
The equation of the function is given as:
y = -16x^2 + 32x
Factor out -16
y = -16(x^2 - 2x)
In the above equation, the factor -16 is negative,
This means that the parabola opens downward
The roots and the x-intercepts
We have:
y = -16(x^2 - 2x)
Set to 0
-16(x^2 - 2x) = 0
Divide by -16
x^2 - 2x = 0
Factor out x
x(x - 2) = 0
Split
x = 0 or x - 2 = 0
Solve for x
x = 0 or x = 2
Hence, the roots are 0 and 2, and the x-intercepts are (0,0) and (2,0)
The x value of the vertex
We have:
y = -16x^2 + 32x
Differentiate
y' = -32x + 32
Set to 0
-32x + 32= 0
Divide through by -32
x - 1 = 0
Add 1 to both sides
x = 1
Hence, the x value of the vertex is 1
The y value of the vertex
In (c), we have:
x = 1
Substitute x = 1 in y = -16x^2 + 32x
y = -16(1)^2 + 32(1)
Evaluate
y = 16
Hence, the y value of the vertex is 16
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