To sketch the graph of the quadratic function h given the characteristics mentioned, we can follow these steps:
Step 1: Identify the equation form
Since the axis of symmetry is the y-axis and the x-intercepts are symmetrical about the y-axis at (-4, 0) and (4, 0), the quadratic function will have a vertex form that meets these characteristics. The general form of such a quadratic equation is \( h(x) = a(x - h)^2 + k \), where (h, k) is the vertex of the parabola.
Step 2: Determine the vertex
Given that the axis of symmetry is the y-axis, the x-coordinate of the vertex must be 0 (since the y-axis is the line x=0). The y-coordinate of the vertex is located at the y-intercept given as -8. So, the vertex of the parabola is (0, -8).
Step 3: Determine the coefficient "a"
Since the x-intercepts are given to be at (-4, 0) and (4, 0), and the vertex is at (0, -8), the function reaches its maximum value at the vertex, meaning that "a" must be negative (the parabola opens downwards). To determine the value of "a", we can use one of the x-intercepts:
\( 0 = a(-4 - 0)^2 + (-8) \) (using the x-intercept (-4, 0))
\( 0 = 16a - 8 \)
Now we solve for a:
\( 16a = 8 \)
\( a = \frac{8}{16} = \frac{1}{2} \)
However, since the parabola opens downwards, a must be negative:
\( a = -\frac{1}{2} \)
Step 4: Write the equation of the quadratic function
Now, with a=-1/2 and the vertex at (0, -8), the equation of the quadratic function is:
\( h(x) = -\frac{1}{2}(x - 0)^2 - 8 \)
Which simplifies to:
\( h(x) = -\frac{1}{2}x^2 - 8 \)
Step 5: Sketch the graph
- Draw a set of axes with x-axis (horizontal) and y-axis (vertical).
- Plot the vertex at (0, -8) and label it.
- Plot and label the x-intercepts at (-4, 0) and (4, 0).
- Since "a" is -1/2, the parabola opens downwards and is a bit wider than the standard parabola \( y = -x^2 \).
- Draw a symmetrical parabolic curve through the vertex and the x-intercepts.
- Make sure to show the intercepts and the axis of symmetry (which will be the y-axis in this case).
Remember to label the axes and the intercepts. The graph should clearly depict a downward-opening parabola with a vertex at (0, -8) and x-intercepts at (-4, 0) and (4, 0).
