Graph the polynomial function. \( \mathrm{h}(\mathrm{x})=\mathrm{x}^{2}+4 \mathrm{x}-5 \) Use the graphing tool to graph the function.

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 \((0,-5)\).