To draw the graph of the quadratic function \( y = x^2 - 5x + 6 \) for the range \( 0 \leq x \leq 5 \), we will follow these steps:
### Step 1: Identify the quadratic function
The given quadratic function is:
\[
y = x^2 - 5x + 6
\]
### Step 2: Factor the quadratic equation
To find the roots (zeros) of the quadratic function, we can factor it:
\[
y = (x - 2)(x - 3)
\]
This gives us the roots:
\[
x = 2 \quad \text{and} \quad x = 3
\]
### Step 3: Calculate the value of \( y \) for various \( x \) values
We will calculate \( y \) for \( x = 0, 1, 2, 3, 4, 5 \):
- For \( x = 0 \):
\[
y = 0^2 - 5(0) + 6 = 6
\]
Point: \( (0, 6) \)
- For \( x = 1 \):
\[
y = 1^2 - 5(1) + 6 = 1 - 5 + 6 = 2
\]
Point: \( (1, 2) \)
- For \( x = 2 \):
\[
y = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0
\]
Point: \( (2, 0) \)
- For \( x = 3 \):
\[
y = 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0
\]
Point: \( (3, 0) \)
- For \( x = 4 \):
\[
y = 4^2 - 5(4) + 6 = 16 - 20 + 6 = 2
\]
Point: \( (4, 2) \)
- For \( x = 5 \):
\[
y = 5^2 - 5(5) + 6 = 25 - 25 + 6 = 6
\]
Point: \( (5, 6) \)
### Step 4: List the points
Now, we have the following points to plot:
- \( (0, 6) \)
- \( (1, 2) \)
- \( (2, 0) \)
- \( (3, 0) \)
- \( (4, 2) \)
- \( (5, 6) \)
### Step 5: Plot the points on a graph
On a Cartesian coordinate system:
- Mark the x-axis from 0 to 5.
- Mark the y-axis from 0 to 6.
- Plot the points calculated above.
### Step 6: Draw the parabola
Connect the points smoothly to form a parabolic curve. The parabola opens upwards, and it intersects the x-axis at \( (2, 0) \) and \( (3, 0) \).
### Step 7: Identify the vertex
The vertex of the parabola can be found using the formula:
\[
x = -\frac{b}{2a} = -\frac{-5}{2 \cdot 1} = \frac{5}{2} = 2.5
\]
Substituting \( x = 2.5 \) into the function to find \( y \):
\[
y = (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25
\]
Thus, the vertex is at \( (2.5, -0.25) \).
