Let's tackle the problem of graphing the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex].
### Step-by-Step Solution
1. Expand the Equation:
Rewrite the quadratic equation in the standard form.
[tex]\[ f(x) = (x-2)(x-6) \][/tex]
Expanding this, we get:
[tex]\[ f(x) = x^2 - 6x - 2x + 12 \][/tex]
[tex]\[ f(x) = x^2 - 8x + 12 \][/tex]
2. Identify the Vertex:
The vertex form of a quadratic equation is [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
For the given quadratic equation [tex]\( y = x^2 - 8x + 12 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 12 \)[/tex]
To find the [tex]\( x \)[/tex]-coordinate of the vertex [tex]\( h \)[/tex], use the formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Substituting the values:
[tex]\[ h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \][/tex]
Now, find the [tex]\( y \)[/tex]-coordinate [tex]\( k \)[/tex] by substituting [tex]\( h = 4 \)[/tex] back into the function:
[tex]\[ k = f(4) = (4-2)(4-6) = 2 \cdot -2 = -4 \][/tex]
So, the vertex of the parabola is [tex]\((4, -4)\)[/tex].
3. Choose Another Point:
Select another point on the parabola for accuracy. We chose [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = (3-2)(3-6) = 1 \cdot -3 = -3 \][/tex]
Therefore, another point on the parabola is [tex]\((3, -3)\)[/tex].
4. Plot the Points:
Use the vertex [tex]\((4, -4)\)[/tex] and the additional point [tex]\((3, -3)\)[/tex] to plot the quadratic function.
To graph [tex]\( f(x) \)[/tex]:
- Plot the vertex at [tex]\((4, -4)\)[/tex].
- Plot the additional point at [tex]\((3, -3)\)[/tex].
- Draw a symmetrical parabola passing through these points, opening upwards since the coefficient of [tex]\( x^2 \)[/tex] is positive (i.e., [tex]\( a = 1 \)[/tex]).
By following these steps, you can accurately graph the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex] using the vertex and another point on the parabola.
