Use the function f(x) = −16x2 + 22x + 3 to answer the questions.

part a: completely factor f(x). (2 points)

part b: what are the x-intercepts of the graph of f(x)? show your work. (2 points)

part c: describe the end behavior of the graph of f(x). explain. (2 points)

part d: what are the steps you would use to graph f(x)? justify that you can use the answers obtained in part b and part c to draw the graph. (4 points)

A) f(x) = -16*(x + 0.125)*(x - 1.5)

B) points (-0.125, 0) and (1.5, 0)

C) tends to negative infinity

D) Draw the line that pass through the roots from item B and has (0.6875, 10.5625) as a vertex.

Step-by-step explanation:

A) To factor a parabola we need to find its roots. We can do that with the quadratic formula:

The factored form is:

f(x) = a*(x -x1)*(x - x2)

then:

f(x) = -16*(x + 0.125)*(x - 1.5)

B) the x-intercepts are its roots, for this case the points (-0.125, 0) and (1.5, 0)

C) Given that the coefficient a of the parabola is negative (-16) then the function open downward. As a consequence its end behaviour tends to negative infinity.

D) To graph f(x) we need 3 points, given that we already have 2 roots, then we need 1 point. One option is calculate the vertex, as follows:

x-component of the vertex of a parabola: -b/(2a)

x-component of the vertex of a parabola: -22/[2(-16)] = 0.6875

The y-component of the vertex of a parabola is obtained replacing the x-component in the formula:

y-component = −16(0.6875)^2 + 22(0.6875) + 3 = 10.5625

So, the vertex coordinate is (0.6875, 10.5625)

A. In order to factor the expression, we need to find its roots: using the quadratic formula, we have

So, the two solutions are

Now, given the polynomial with roots , its factorization is

In this case, we have

B. The x-intercepts are the roots we just found:

C. The end behaviour of a parabola is

Meaning that the limits are both positive infinity if a is positive, and negative infinity if a is negative. In this case, a is -16, and thus we have

D. We already have almost all the information we need. You just need to find out the vertex of the parabola, and then you could sketch the graph. The vertex is a maximum because the parabola is facing downwards, and from the vertex you can descend towards the roots (part B), and continue symmetrically towards negative infinity (part C).

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step-by-step explanation: