To sketch the graph of the equation [tex]\( y = (x-3)^2 - 16 \)[/tex], follow these steps:
1. Identify the Basic Shape:
The given equation [tex]\( y = (x-3)^2 - 16 \)[/tex] is written in the form [tex]\( y = a(x-h)^2 + k \)[/tex], which represents a parabola. Since the coefficient [tex]\( a \)[/tex] (the number in front of the squared term) is positive (implicitly 1 here), the parabola opens upwards.
2. Find the Vertex:
The vertex form of the quadratic equation [tex]\( y = a(x-h)^2 + k \)[/tex] reveals that the vertex is at [tex]\( (h, k) \)[/tex].
Here, [tex]\( h = 3 \)[/tex] and [tex]\( k = -16 \)[/tex], so the vertex of the parabola is at the point [tex]\( (3, -16) \)[/tex].
3. Determine Additional Points:
To get a sense of the parabola’s shape, calculate the y-values for several x-values on each side of the vertex.
- For [tex]\( x = 0 \)[/tex], [tex]\( y = (0-3)^2 - 16 = 9 - 16 = -7 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( y = (1-3)^2 - 16 = 4 - 16 = -12 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = (2-3)^2 - 16 = 1 - 16 = -15 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = (4-3)^2 - 16 = 1 - 16 = -15 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y = (5-3)^2 - 16 = 4 - 16 = -12 \)[/tex]
- For [tex]\( x = 6 \)[/tex], [tex]\( y = (6-3)^2 - 16 = 9 - 16 = -7 \)[/tex]
This provides the points:
[tex]\[
(0, -7), (1, -12), (2, -15), (4, -15), (5, -12), (6, -7)
\][/tex]
4. Plot the Points:
Plot these points on a coordinate plane along with the vertex (3, -16):
- Vertex: [tex]\( (3, -16) \)[/tex]
- Additional points: [tex]\( (0, -7), (1, -12), (2, -15), (4, -15), (5, -12), (6, -7) \)[/tex]
5. Draw the Parabola:
Connect these points smoothly to form a U-shaped curve opening upwards.
6. Analyze Graph Choices:
Compare your sketched graph with the provided options (A, B, C, D) to determine which one matches the characteristics of your parabola.
The graph of the equation [tex]\( y = (x-3)^2 - 16 \)[/tex] should show a U-shaped curve that opens upwards with the vertex at [tex]\( (3, -16) \)[/tex] and passing through the points discussed.
Select the graph that best matches these characteristics:
- Vertex at [tex]\( (3, -16) \)[/tex]
- Symmetric points around the vertex at [tex]\( (0, -7), (1, -12), (2, -15), (4, -15), (5, -12), (6, -7) \)[/tex].
