Let's go through the steps to graph the quadratic function with a vertex at (0, 4) and x-intercepts at (-2, 0) and (2, 0) in a detailed manner.
### Step-by-Step Solution:
1. Identify Key Points:
- Vertex: The vertex is given as [tex]\((0, 4)\)[/tex].
- X-intercepts: The x-intercepts are given as [tex]\((-2, 0)\)[/tex] and [tex]\((2, 0)\)[/tex].
2. Plot the Points:
- First, plot the vertex [tex]\((0, 4)\)[/tex] on the graph.
- Next, plot the x-intercept points [tex]\((-2, 0)\)[/tex] and [tex]\((2, 0)\)[/tex] on the graph.
These points are crucial as they define key characteristics of the quadratic function.
3. Graph Symmetry:
- The graph of a quadratic function is always symmetrical about the vertical line that passes through the vertex. In this case, the vertex is at [tex]\((0, 4)\)[/tex], so the line of symmetry is the y-axis or [tex]\(z = 0\)[/tex].
4. Connect Points with Curved Lines:
- Since it is a quadratic function (parabola), after plotting the vertex and x-intercepts, connect these points using a smooth, curved line that opens downward. The parabola will intersect the x-axis at the given intercepts and have its maximum point at the vertex.
5. Find the Slope:
- To find the slope between the vertex [tex]\((0, 4)\)[/tex] and one of the x-intercepts (let’s choose [tex]\((-2, 0)\)[/tex]):
- Use the formula [tex]\( \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex]
- Here, [tex]\((x_1, y_1) = (0, 4)\)[/tex] and [tex]\((x_2, y_2) = (-2, 0)\)[/tex]
- Substitute the values:
[tex]\[
\text{slope} = \frac{0 - 4}{-2 - 0} = \frac{-4}{-2} = 2
\][/tex]
The slope between the vertex [tex]\((0, 4)\)[/tex] and the x-intercept [tex]\((-2, 0)\)[/tex] is 2.0.
### Summary:
- Vertex: (0, 4)
- X-intercepts: (-2, 0) and (2, 0)
- Slope between vertex and one x-intercept: 2.0
After plotting these points, ensure you draw a smooth, curved line that forms a parabola, symmetrical about [tex]\(z = 0\)[/tex], with its highest point at the vertex [tex]\((0, 4)\)[/tex].
