Answer
To graph the polynomial function \( h(x) = x^2 + 4x - 5 \):
1. **Identify the type:** It's a quadratic function (parabola) opening upward.
2. **Find roots:** \( x = -5 \) and \( x = 1 \).
3. **Find vertex:** At \( x = -2 \), \( h(-2) = -9 \).
4. **Find y-intercept:** At \( x = 0 \), \( h(0) = -5 \).
5. **Plot points:** Vertex \((-2, -9)\), roots \((-5, 0)\) and \((1, 0)\), and y-intercept \((0, -5)\).
6. **Draw the parabola:** Sketch the upward-opening curve symmetric around \( x = -2 \) passing through these points.
Use a graphing tool to plot these points and draw the parabola accordingly.
Solution
1. **Identify the function and its type:**
The function is
\[
\mathrm{h}(x)=x^2+4x-5.
\]
It is a quadratic function (a parabola) that opens upward because the coefficient of \(x^2\) is positive.
2. **Find the roots (x-intercepts):**
Solve
\[
x^2+4x-5=0.
\]
Factor the quadratic:
\[
x^2+4x-5=(x+5)(x-1)=0.
\]
Setting each factor equal to zero:
\[
x+5=0\quad \text{or} \quad x-1=0,
\]
which gives the roots
\[
x=-5 \quad \text{and} \quad x=1.
\]
3. **Find the vertex:**
The x-coordinate of the vertex is given by
\[
x=-\frac{b}{2a}=-\frac{4}{2(1)}=-2.
\]
Substitute \(x=-2\) into \(\mathrm{h}(x)\) to find the y-coordinate:
\[
\mathrm{h}(-2)=(-2)^2+4(-2)-5=4-8-5=-9.
\]
Therefore, the vertex is
\[
(-2,-9).
\]
4. **Determine the y-intercept:**
The y-intercept is found by evaluating the function at \(x=0\):
\[
\mathrm{h}(0)=0^2+4(0)-5=-5.
\]
So, the y-intercept is \((0,-5)\).
5. **Graphing instructions:**
- **Plot the vertex:** Plot the point \((-2,-9)\).
- **Plot the roots (x-intercepts):** Plot the points \((-5,0)\) and \((1,0)\).
- **Plot the y-intercept:** Plot the point \((0,-5)\).
- **Axis of symmetry:** Draw a vertical line through \(x=-2\).
- **Draw the parabola:** Sketch the curve opening upward, making sure it is symmetric with respect to the line \(x=-2\) and passes through all the key points.
6. **Use the graphing tool:**
Enter the function
\[
\mathrm{h}(x)=x^2+4x-5
\]
into your graphing tool to verify the shape of the parabola and the plotted points.
The final graph will be an upward-opening parabola with vertex at \((-2,-9)\), x-intercepts at \((-5,0)\) and \((1,0)\), and y-intercept at \((0,-5)\).
Answered by UpStudy AI and reviewed by a Professional Tutor
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