To graph the equation [tex]\( y = x^2 - 12x + 32 \)[/tex] and plot 5 key points including the roots and the vertex, we need to follow these steps:
### Step 1: Identify the Quadratic Equation
The quadratic equation given is:
[tex]\[ y = x^2 - 12x + 32 \][/tex]
### Step 2: Find the Roots
The roots of the quadratic equation are the values of [tex]\( x \)[/tex] where [tex]\( y = 0 \)[/tex].
The roots are [tex]\( (8, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
### Step 3: Find the Vertex
The vertex form of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
For this equation, [tex]\( a = 1 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 32 \)[/tex]. Thus,
[tex]\[ x_{\text{vertex}} = -\frac{-12}{2 \cdot 1} = 6 \][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 6 \)[/tex] back into the equation:
[tex]\[ y_{\text{vertex}} = 6^2 - 12 \cdot 6 + 32 \][/tex]
[tex]\[ y_{\text{vertex}} = 36 - 72 + 32 \][/tex]
[tex]\[ y_{\text{vertex}} = -4 \][/tex]
So, the vertex is [tex]\( (6, -4) \)[/tex].
### Step 4: Identify Two Additional Points
For a thorough graph, it's helpful to include two additional points. We can use the y-intercept (when [tex]\( x = 0 \)[/tex]) and another standard point, for instance, [tex]\( x = 12 \)[/tex].
1. [tex]\( (0, 32) \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0^2 - 12 \cdot 0 + 32 \][/tex]
[tex]\[ y = 32 \][/tex]
So, the point is [tex]\( (0, 32) \)[/tex].
2. [tex]\( (12, 32) \)[/tex]:
Substitute [tex]\( x = 12 \)[/tex]:
[tex]\[ y = 12^2 - 12 \cdot 12 + 32 \][/tex]
[tex]\[ y = 144 - 144 + 32 \][/tex]
[tex]\[ y = 32 \][/tex]
So, the point is [tex]\( (12, 32) \)[/tex].
### Step 5: Plot the Points
Plot the five points:
1. The roots: [tex]\( (8, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex]
2. The vertex: [tex]\( (6, -4) \)[/tex]
3. The y-intercept: [tex]\( (0, 32) \)[/tex]
4. Another point: [tex]\( (12, 32) \)[/tex]
### Step 6: Draw the Parabola
Using these points, we can draw the parabola. The graph opens upwards because the coefficient of [tex]\( x^2 \)[/tex] (which is 1) is positive.
In summary, the 5 key points to plot on the graph of [tex]\( y = x^2 - 12x + 32 \)[/tex] are:
1. [tex]\( (8, 0) \)[/tex]
2. [tex]\( (4, 0) \)[/tex]
3. [tex]\( (6, -4) \)[/tex]
4. [tex]\( (0, 32) \)[/tex]
5. [tex]\( (12, 32) \)[/tex]
By plotting these points, you can draw an accurate graph of the quadratic function.
