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Graph the polynomial function.
h(x)= x^2+4x-5

User Tammara
by
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2 votes

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).

User Azee
by
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