Partial Factoring involves finding two points on the parabola that have the same y-coordinate. We know that if we have two points on a parabola that have the same y-coordinate, we can find the x-coordinate of the vertex by taking the average of the two coordinates.
Steps to follow when Partial Factoring.
1. Look at non x term in the expression. Take away that amount from each side of the expression.
2. Factor whatever is left in the expression, and find the two places where x is equal to zero. Note: These are not the roots. This x-value represents where the y- coordinates are the same.
3. Find the Axis of Symmetry of the parabola. We can do this by taking the average of the two x-coordinates where the y-values are the same. Now, we know the x-value of the vertex.
4. Substitute the x-value of the vertex into the original equation to find out the y-value of the vertex.
Example 1: Use Partial Factoring to find the vertex of the parabola with the equation y = x² + 2x - 35
y = x² + 2x 35
Let y = -35
-35 = x² + 2x 35 [if we rearrange the expression, and move the left side over to the right side, we will get the following expression:]
0 = x² + 2x [it is easy to now factor this expression]
0 = x (x + 2) [now, solve for x]
x = 0 or x = -2.
When the parabola has a y-coordinate of 35, the matching x-coordinates are 0 and 2.
Now, find the average of the two x-coordinates to find the Axis of Symmetry of the parabola.
(0 + (-2)) χ 2 = -1
Therefore, the vertex is at the point (-1,y)
We now have to find out what the matching y-coordinate is for the vertex.
When x = -1,
y = (-1)² + 2(-1) 35
y = 1 2 35
y = -36
Therefore, the vertex is at the point (-1,-36)
Practice Questions
Page 377 #5, 6, 7, 10
