Answer:
Explanation:
To graph the polynomial function \(h(x) = x^2 + 4x - 5\), you can follow these steps:
1. Find the x-intercepts (if any):
To find the x-intercepts, set \(h(x) = 0\) and solve for \(x\):
\(x^2 + 4x - 5 = 0\)
This is a quadratic equation. You can factor it or use the quadratic formula. Factoring it, you get:
\((x + 5)(x - 1) = 0\)
So, \(x = -5\) or \(x = 1\). These are the x-intercepts.
2. Find the y-intercept:
To find the y-intercept, set \(x = 0\) in the equation:
\(h(0) = 0^2 + 4(0) - 5 = -5\)
So, the y-intercept is -5, which gives you the point (0, -5).
3. Find the vertex:
The vertex of the parabola is given by the formula \(x = -\frac{b}{2a}\), where \(a\) and \(b\) are the coefficients of the quadratic term and the linear term in the equation, respectively.
In this case, \(a = 1\) and \(b = 4\). So, the x-coordinate of the vertex is:
\(x = -\frac{4}{2(1)} = -2\)
To find the corresponding y-coordinate, plug this x-value into the original equation:
(h(-2) = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9
)
So, the vertex is (-2, -9).
4. Determine the axis of symmetry:
The axis of symmetry is a vertical line that goes through the vertex. In this case, it's the line (x = -2).
5. Plot the points:
Plot the x-intercepts (-5, 0) and (1, 0), the y-intercept (0, -5), and the vertex (-2, -9).
6. Determine the direction of the parabola:
Since the coefficient of the (x^2) term is positive (1), the parabola opens upward.
7. Draw the graph:
Based on the points you've plotted and the direction of the parabola, you can draw a smooth U-shaped curve that passes through these points, with the axis of symmetry at (x = -2).
Here's a rough sketch of the graph:
```
2
|
|
|
| 1
|
|
|
|
0
|---|---|---|---|---|---|---|---|---|
-7 -6 -5 -4 -3 -2 -1 0 1
```
The parabola opens upward, and the vertex is at (-2, -9). The x-intercepts are at -5 and 1, and the y-intercept is at -5. The axis of symmetry is the vertical line(x = -2).