20 points! Use the function f(x) to answer the questions. f(x) = −16x2 + 60x + 16 Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points) Part B: Is the vertex of the graph of f(x) going to be a maximum or a minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points) Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

See below.

Step-by-step explanation:

f(x) = −16x2 + 60x + 16

A:

Set the function equal to zero and solve for x.

−16x2 + 60x + 16 = 0

4x^2 - 15x - 4 = 0

(4x + 1)(x - 4) = 0

4x + 1 = 0 or x - 4 = 0

4x = -1 or x = 4

x = -1/4 or x = 4

The x-intercepts are x = -1/4 and x = 4.

B:

Since the leading term, -16x^2, has a negative coefficient, the graph of this function is a parabola that opens downward. The vertex is the maximum value of the function.

Because of symmetry, the vertex occurs at a value of x which is midway between the x-intercepts.

x = (-1/4 + 4)/2 = (15/4)/2 = 15/8 = 1.875

Now that we know the x-coordinate of the vertex, we plug it in the function to find the y-coordinate.

f(1.875) = −16(1.875)^2 + 60(1.875) + 16

f(1.875) = 72.25

The coordinates of the vertex are (1.875, 72.25)

C:

First, graph the vertex, point (1.875, 72.25).

Then graph the x-intercepts, (-0.25, 0) and (4, 0).

To help graph, you can get a few more points on both sides of the vertex by evaluating the function at 0.875, 2.875 and -3.125, 6.875, or other points you may prefer. Then draw the curve of the parabola passing through those points.

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step-by-step explanation: