To graph the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex] using the parabola tool, follow these steps:
1. Find the vertex of the parabola:
- To convert the function [tex]\( f(x) = (x-2)(x-6) \)[/tex] to its standard form [tex]\( ax^2 + bx + c \)[/tex], expand the expression:
[tex]\[
(x-2)(x-6) = x^2 - 6x - 2x + 12 = x^2 - 8x + 12
\][/tex]
- The standard form is [tex]\( f(x) = x^2 - 8x + 12 \)[/tex].
- The vertex of a parabola in the form [tex]\( ax^2 + bx + c \)[/tex] is given by the formula [tex]\(\left( -\frac{b}{2a}, f\left( -\frac{b}{2a} \right) \right)\)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 12 \)[/tex].
- Calculate the x-coordinate of the vertex:
[tex]\[
h = -\frac{b}{2a} = -\frac{-8}{2 \times 1} = \frac{8}{2} = 4
\][/tex]
- Calculate the y-coordinate by substituting [tex]\( x = 4 \)[/tex] back into the quadratic function:
[tex]\[
k = f(4) = 4^2 - 8 \times 4 + 12 = 16 - 32 + 12 = -4
\][/tex]
- Therefore, the vertex of the parabola is [tex]\( (4, -4) \)[/tex].
2. Plot the vertex:
- Locate the point [tex]\( (4, -4) \)[/tex] on the graph and plot it.
3. Find a second point on the parabola:
- Choose another value of [tex]\( x \)[/tex] to find a corresponding [tex]\( y \)[/tex]-value. Let's select [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0^2 - 8 \times 0 + 12 = 12
\][/tex]
- Thus, the point is [tex]\( (0, 12) \)[/tex].
4. Plot the second point:
- Locate the point [tex]\( (0, 12) \)[/tex] on the graph and plot it.
5. Draw the parabola:
- Use the parabola tool to draw a smooth curve through the points [tex]\( (4, -4) \)[/tex] and [tex]\( (0, 12) \)[/tex], making sure to shape it correctly to reflect the symmetry of a parabola.
By following these steps, you can accurately graph the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex] on the coordinate plane.
