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.
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