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.

for this case we have by definition, if two lines are perpendicular, then the product of their slopes is -1.

then, given the following line:

so,

we are looking for m_ {2}:

then, the line is given by:

we find "b" replacing the given point:

finally, the equation is:

option c