To graph the equation [tex]\( y = 5(x-1)^2 - 5 \)[/tex], we can follow these steps:
1. Identify the Basic Structure: The equation is in the form [tex]\( y = a(x-h)^2 + k \)[/tex], which results in a parabola. Here, [tex]\( a = 5 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = -5 \)[/tex]. The vertex form of a parabola reveals the vertex and direction of the parabola. The vertex of this parabola is at [tex]\( (h, k) = (1, -5) \)[/tex].
2. Determine the Opening Direction: Since the coefficient of the squared term, [tex]\( a \)[/tex], is positive (i.e., [tex]\( a = 5 \)[/tex]), the parabola opens upwards.
3. Vertex and Axes: The vertex is a crucial point at [tex]\( (1, -5) \)[/tex]. The axis of symmetry is the vertical line that passes through the vertex, which is [tex]\( x = 1 \)[/tex].
4. Calculate Key Points: To plot the parabola accurately, calculate additional points on either side of the vertex.
- Let [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 5(0-1)^2 - 5 = 5(1) - 5 = 0
\][/tex]
So, [tex]\( (0, 0) \)[/tex] is a point on the graph.
- Let [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 5(2-1)^2 - 5 = 5(1) - 5 = 0
\][/tex]
So, [tex]\( (2, 0) \)[/tex] is another point on the graph.
- Let [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 5(-1-1)^2 - 5 = 5(4) - 5 = 20 - 5 = 15
\][/tex]
Therefore, [tex]\( (-1, 15) \)[/tex] is another point.
- Let [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 5(3-1)^2 - 5 = 5(4) - 5 = 20 - 5 = 15
\][/tex]
Hence, [tex]\( (3, 15) \)[/tex] is another point.
5. Plot the points and Sketch the Curve:
- Start at the vertex [tex]\((1, -5)\)[/tex].
- Plot the calculated points: [tex]\((0, 0)\)[/tex], [tex]\((2, 0)\)[/tex], [tex]\((-1, 15)\)[/tex], and [tex]\((3, 15)\)[/tex].
- Draw a smooth, U-shaped curve through these points, ensuring the axis of symmetry is [tex]\(x=1\)[/tex].
6. Draw Axes and Label: On your graph paper or software, draw the x-axis and y-axis. Make sure to clearly label the vertex and the points and draw the curve passing through these points.
### Graph Overview:
- Vertex: (1, -5)
- Axis of Symmetry: x = 1
- Direction: Opens upward because the coefficient of [tex]\( (x-1)^2 \)[/tex] is positive
- Important Points: (0, 0), (2, 0), (-1, 15), (3, 15)
This completes the graph of the given quadratic function [tex]\( y = 5(x-1)^2 - 5 \)[/tex].
