To graph the function \( y = -\frac{1}{2}x^2 + 4 \), we can follow these steps:
Step 1: Identify the shape of the graph
The function is a quadratic equation in the form \( y = ax^2 + bx + c \). Here, \( a = -\frac{1}{2} \), meaning the parabola opens downward.
Step 2: Find the vertex
The vertex form of a parabola is obtained from the standard form by completing the square or can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]
In this function, \( b = 0 \) (since there is no \( x \) term) and \( a = -\frac{1}{2} \):
\[ x = -\frac{0}{2 \times -\frac{1}{2}} = 0 \]
Now, substitute \( x = 0 \) back into the equation to find \( y \):
\[ y = -\frac{1}{2}(0)^2 + 4 = 4 \]
Thus, the vertex of the parabola is \( (0, 4) \).
Step 3: Find the x-intercepts
To find the x-intercepts, set \( y = 0 \):
\[ 0 = -\frac{1}{2}x^2 + 4 \]
Rearranging gives:
\[ \frac{1}{2}x^2 = 4 \] \[ x^2 = 8 \] \[ x = \pm \sqrt{8} = \pm 2\sqrt{2} \]
The x-intercepts are approximately \( x \approx 2.83 \) and \( x \approx -2.83 \).
Step 4: Find the y-intercept
The y-intercept occurs when \( x = 0 \):
\[ y = -\frac{1}{2}(0)^2 + 4 = 4 \]
Thus, the y-intercept is at \( (0, 4) \).
Step 5: Select additional points
To get a better shape of the parabola, we can choose a few values around the vertex to find corresponding y-values:
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For \( x = 2 \): \[ y = -\frac{1}{2}(2^2) + 4 = -\frac{1}{2}(4) + 4 = -2 + 4 = 2 \]
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For \( x = -2 \): \[ y = -\frac{1}{2}(-2^2) + 4 = -\frac{1}{2}(4) + 4 = -2 + 4 = 2 \]
Step 6: Graph the points
You can plot the following points:
- Vertex: \( (0, 4) \)
- X-intercepts: \( (2\sqrt{2}, 0) \) and \( (-2\sqrt{2}, 0) \) (approximately \( (2.83, 0) \) and \( (-2.83, 0) \))
- Y-intercept: \( (0, 4) \)
- Additional points: \( (2, 2) \) and \( (-2, 2) \)
Step 7: Draw the parabola
Using these points, sketch the graph of the parabola, making sure it opens downward. The graph will have a symmetrical shape about the y-axis due to the absence of an x term.
If you have graphing software or a graphing calculator, you can input the equation directly to visualize the parabola accurately. However, the key points and features mentioned will guide you in manually sketching the graph.
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