To graph the quadratic function [tex]\( f(x) = (x-3)^2 \)[/tex], let's follow these steps:
1. Identify the Vertex and Axis of Symmetry:
- The function [tex]\( f(x) = (x-3)^2 \)[/tex] is in vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
- Here, [tex]\( h = 3 \)[/tex] and [tex]\( k = 0 \)[/tex], so the vertex of the parabola is [tex]\((3, 0)\)[/tex].
- The axis of symmetry is the vertical line [tex]\( x = 3 \)[/tex].
2. Determine the Direction of the Parabola:
- The coefficient [tex]\( a \)[/tex] in the vertex form [tex]\( a(x-h)^2 + k \)[/tex] determines the direction of the parabola.
- Since [tex]\( a = 1 \)[/tex], which is positive, the parabola opens upwards.
3. Calculate Key Points:
- Choose a few [tex]\( x \)[/tex]-values around the vertex and calculate the corresponding [tex]\( y \)[/tex]-values.
4. Plot the Points and Draw the Parabola:
- After calculating the key points, plot them on the coordinate grid and draw a smooth curve through these points to represent the parabola.
Let's calculate some key points:
- At [tex]\( x = -10 \)[/tex]:
[tex]\[
f(-10) = (-10 - 3)^2 = (-13)^2 = 169
\][/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = (0 - 3)^2 = (-3)^2 = 9
\][/tex]
- At [tex]\( x = 3 \)[/tex] (the vertex):
[tex]\[
f(3) = (3 - 3)^2 = (0)^2 = 0
\][/tex]
- At [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = (6 - 3)^2 = (3)^2 = 9
\][/tex]
- At [tex]\( x = 10 \)[/tex]:
[tex]\[
f(10) = (10 - 3)^2 = (7)^2 = 49
\][/tex]
These points will help us draw the parabola accurately.
### Graphing the Points:
1. Plot the vertex: [tex]\((3, 0)\)[/tex].
2. Plot additional points [tex]\((-10, 169)\)[/tex], [tex]\((0, 9)\)[/tex], [tex]\((6, 9)\)[/tex], and [tex]\((10, 49)\)[/tex].
Now, let's connect these points with a smooth curve to form the parabola.
### Important Properties of the Graph:
- Vertex: [tex]\((3, 0)\)[/tex]
- Axis of Symmetry: [tex]\( x = 3 \)[/tex]
- Direction: Opens upwards
By following these steps, you can accurately draw the graph of the function [tex]\( f(x) = (x-3)^2 \)[/tex].
