To graph the parabola given by the equation [tex]\( y = x^2 - 10x + 22 \)[/tex], we need to find the vertex and four additional points to accurately plot the curve. Here are the step-by-step calculations and the points required:
### Step 1: Identify the Coefficients
The quadratic equation is in the form [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -10 \)[/tex]
- [tex]\( c = 22 \)[/tex]
### Step 2: Calculate the Vertex
The vertex [tex]\((h, k)\)[/tex] of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the following formulas:
- [tex]\( h = -\frac{b}{2a} \)[/tex]
- [tex]\( k = c - \frac{b^2}{4a} \)[/tex]
Substituting the values:
- [tex]\( h = -\frac{-10}{2 \cdot 1} = 5.0 \)[/tex]
- [tex]\( k = 22 - \frac{(-10)^2}{4 \cdot 1} = 22 - 25 = -3.0 \)[/tex]
So, the vertex of the parabola is [tex]\((5.0, -3.0)\)[/tex].
### Step 3: Find Additional Points
To better define the shape of the parabola, we need four additional points, two on each side of the vertex. We can calculate these by selecting [tex]\(x\)[/tex]-values close to [tex]\(h = 5\)[/tex] and finding the corresponding [tex]\(y\)[/tex]-values.
#### Points to the Left of the Vertex (x = h - 1 and x = h - 2)
1. When [tex]\( x = 4.0 \)[/tex]:
[tex]\[
y = (4)^2 - 10(4) + 22 = 16 - 40 + 22 = -2.0
\][/tex]
So, one point is [tex]\((4.0, -2.0)\)[/tex].
2. When [tex]\( x = 3.0 \)[/tex]:
[tex]\[
y = (3)^2 - 10(3) + 22 = 9 - 30 + 22 = 1.0
\][/tex]
So, another point is [tex]\((3.0, 1.0)\)[/tex].
#### Points to the Right of the Vertex (x = h + 1 and x = h + 2)
3. When [tex]\( x = 6.0 \)[/tex]:
[tex]\[
y = (6)^2 - 10(6) + 22 = 36 - 60 + 22 = -2.0
\][/tex]
So, another point is [tex]\((6.0, -2.0)\)[/tex].
4. When [tex]\( x = 7.0 \)[/tex]:
[tex]\[
y = (7)^2 - 10(7) + 22 = 49 - 70 + 22 = 1.0
\][/tex]
So, the last point is [tex]\((7.0, 1.0)\)[/tex].
### Summary of Points
- Vertex: [tex]\( (5.0, -3.0) \)[/tex]
- Additional Points:
- [tex]\((4.0, -2.0)\)[/tex]
- [tex]\((3.0, 1.0)\)[/tex]
- [tex]\((6.0, -2.0)\)[/tex]
- [tex]\((7.0, 1.0)\)[/tex]
### Plotting the Parabola
With these points [tex]\((5.0, -3.0) \)[/tex], [tex]\((4.0, -2.0) \)[/tex], [tex]\((3.0, 1.0) \)[/tex], [tex]\((6.0, -2.0) \)[/tex], and [tex]\((7.0, 1.0) \)[/tex], you can now accurately plot the parabola [tex]\( y = x^2 - 10x + 22 \)[/tex]. Mark these points on the coordinate plane and draw a smooth curve through them to represent the shape of the parabola.
