Question

Graph the function f(x) = (x + 1)(x – 5). Use the drop-down menus to complete the steps needed to graph the function.

Identify the x-intercepts: (–1, 0) and (5, 0)
Find the midpoint between the intercepts (calculate, don't plot).
Find the vertex:
Find the y-intercept:
Plot another point, then draw the graph.

Answer 1

To graph the function \( f(x) = (x + 1)(x - 5) \), follow these steps:

### 1. Identify the x-intercepts:
The x-intercepts occur where \( f(x) = 0 \). Setting the function to zero gives:
\[
(x + 1)(x - 5) = 0
\]
This yields:
- \( x + 1 = 0 \) → \( x = -1 \) (Intercept: (-1, 0))
- \( x - 5 = 0 \) → \( x = 5 \) (Intercept: (5, 0))

**X-intercepts identified:** \( (-1, 0) \) and \( (5, 0) \)

### 2. Find the midpoint between the intercepts:
To find the midpoint between the x-intercepts \(-1\) and \(5\):
\[
\text{Midpoint} = \left( \frac{-1 + 5}{2}, \frac{0 + 0}{2} \right) = \left( \frac{4}{2}, 0 \right) = (2, 0)
\]

### 3. Find the vertex:
The vertex of a parabola in standard form can be found at the average of the x-intercepts:
\[
x = 2
\]
Now, substitute \( x = 2 \) back into \( f(x) \) to find the y-coordinate of the vertex:
\[
f(2) = (2 + 1)(2 - 5) = 3 \cdot (-3) = -9
\]

**Vertex identified:** \( (2, -9) \)

### 4. Find the y-intercept:
The y-intercept occurs where \( x = 0 \):
\[
f(0) = (0 + 1)(0 - 5) = 1 \cdot (-5) = -5
\]

**Y-intercept identified:** \( (0, -5) \)

### 5. Plot another point:
To plot another point, we can pick another value of \( x \). Let's choose \( x = 3 \):
\[
f(3) = (3 + 1)(3 - 5) = 4 \cdot (-2) = -8
\]
This gives the point \( (3, -8) \).

### 6. Draw the graph:
Now, we can plot the points:

- X-intercepts: \( (-1, 0) \) and \( (5, 0) \)
- Y-intercept: \( (0, -5) \)
- Vertex: \( (2, -9) \)
- Additional point: \( (3, -8) \)

Drawing a smooth curve through these points will yield the graph of the parabola. The parabola opens upward since the coefficient of \( x^2 \) in the expanded form \( f(x) = x^2 - 4x - 5 \) is positive.