Question

07.03, 07.05 HC)

Use the function f(x) = −16×2 + 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 dra

1. 1. The x-intercepts of the graph of f(x) are (-1/8, 3/2).
2. The parabola opens downwards, the vertex is a maximum and this is simply because the coefficient of x² is negative.
3. Mark the roots of the quadratic function on the x-axis (-1/8, 3/2).
4. Determine the y-intercept (3).
5. Connect all the known points on the graph to form a downward parabola.

### What is a quadratic function?

A quadratic function can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.
In Mathematics, the graph of any quadratic function is parabolic because it is a u-shaped curve. For the given quadratic function, the graph is a downward parabola because the coefficient of x² is negative.

### How to determine the x-intercepts?

In order to determine the x-intercepts of the graph of f(x), we would make to be y = 0.
By rearranging and completely factoring the given quadratic function, we have:
f(x) = -16x² + 22x + 3
f(x) = 16x² – 22x – 3
16x² – 24x + 2x – 3 = 0
8x(2x – 3) + 1(2x – 3) = 0
(8x + 1)(2x – 3) = 0
x₁ ⇒ 8x = -1      ⇒     x₁ = -1/8.
x₂ ⇒ 2x = 3      ⇒     x₂ = 3/2.
Therefore, the x-intercepts of the graph of f(x) are (-1/8, 3/2).

### How to describe the end behavior?

Since the parabola opens downwards, the vertex is a maximum and this is simply because the coefficient of x² is negative.
V(x) = -b/2a
V(x) = -22/2(-16)
V(x) = -22/-32
V(x) = 11/16
V(y) = -D/4a
V(y) = -16(11/16)² + 22(11/16) + 3
V(y) = -16(121/256) + 121/8 + 3
V(y) = -121/16 + 121/8 + 3
V(y) = (-121 + 242 + 48)/16
V(y) = 169/16.

### What are the steps you would use to graph f(x)?

• Mark the roots of the quadratic function on the x-axis (-1/8, 3/2).
• Determine the y-intercept (3).
• Connect all the known points on the graph to form a downward parabola.