To graph the quadratic equation [tex]\( y = x^2 - 6x + 5 \)[/tex], you need to find and plot specific points, including the roots (where the graph intersects the x-axis), the vertex (the highest or lowest point on the graph), and two additional points to better illustrate the curve. Here are the steps to achieve this:
### Step 1: Find the Roots
The roots can be found by solving the equation [tex]\( y = 0 \)[/tex]:
[tex]\[ x^2 - 6x + 5 = 0 \][/tex]
This quadratic equation can be factored into:
[tex]\[ (x - 1)(x - 5) = 0 \][/tex]
The solutions (roots) are:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = 5 \][/tex]
So, the roots are at the points:
[tex]\[ (1, 0) \][/tex]
[tex]\[ (5, 0) \][/tex]
### Step 2: Find the Vertex
The vertex of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For this equation, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex]:
[tex]\[ x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \][/tex]
To find the y-coordinate, substitute [tex]\( x = 3 \)[/tex] back into the equation:
[tex]\[ y = (3)^2 - 6(3) + 5 \][/tex]
[tex]\[ y = 9 - 18 + 5 \][/tex]
[tex]\[ y = -4 \][/tex]
So, the vertex is at the point:
[tex]\[ (3, -4) \][/tex]
### Step 3: Find Two Additional Points
To plot the graph more accurately, choose two additional points. A good choice is usually to take x-values on either side of the vertex but different from the roots. We can choose [tex]\( x = 0 \)[/tex] and [tex]\( x = 6 \)[/tex]:
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = (0)^2 - 6(0) + 5 = 5 \][/tex]
So, the point is:
[tex]\[ (0, 5) \][/tex]
For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = (6)^2 - 6(6) + 5 \][/tex]
[tex]\[ y = 36 - 36 + 5 = 5 \][/tex]
So, the point is:
[tex]\[ (6, 5) \][/tex]
### Step 4: Plot the Points and Draw the Graph
The points to plot are:
- Roots: [tex]\((1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex]
- Vertex: [tex]\((3, -4)\)[/tex]
- Additional Points: [tex]\((0, 5)\)[/tex] and [tex]\((6, 5)\)[/tex]
On a graph, plot these points and then draw a smooth curve through them to visualize the graph of the quadratic equation [tex]\( y = x^2 - 6x + 5 \)[/tex].
The graph should look like a parabola opening upwards, with the vertex at [tex]\((3, -4)\)[/tex] and crossing the x-axis at [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
