x |
y |
Critical Attribute |
|---|---|---|
-2 |
10 |
|
-1 |
0 |
x-intercept |
0 |
-6 |
y-intercept |
1 |
-8 |
Vertex (minimum) |
2 |
-6 |
Symmetric point |
3 |
0 |
x-intercept |
4 |
10 |
x |
y |
Critical Attribute |
|---|---|---|
-2 |
10 |
|
-1 |
0 |
x-intercept |
0 |
-6 |
y-intercept |
1 |
-8 |
Vertex (minimum) |
2 |
-6 |
Symmetric point |
3 |
0 |
x-intercept |
4 |
10 |
To write an equation, you will actually use the coordinates of the points observed from the graph. The points are displayed in the table.
Using the coordinates of the x-intercepts, you can write an equation in factored form with the coefficient of x2 to be found:
Since the zeros (x-intercepts) occur when x = -1 and x = 3,
y = a(x + 1)(x - 3)
Now, substitute the coordinates from one of the other points on the graph, such as (-2, 10):
Note x-values are indicated in red and y-values are indicated in green.
And solve for “a”
10 = a( -2 + 1 )( -2 - 3 )
10 = a( -1 )( -5 )
10 = 5a
2 = a
Now, substitute the value for a back into your equation with your variables:
y = 2(x + 1)(x − 3)
y = 2(x2 − 3x + 1x − 3)
Multiply the two binomials together by distribution.
y = 2(x2 −2x − 3)
Combine the x–terms within the parentheses.
y = 2x2 − 4x − 6
Distribute the 2 to all terms and simplify.