The Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction.

• If the parabola opens up, the vertex is the lowest point on the graph. This is called the minimum. • If the parabola opens down, the vertex is the highest point on the graph. This is called the maximum.

If you fold the graph along the axis of symmetry one side of the graph will lie on top of the other.

Step 1 Find the x-coordinate. It is given by the formula .

Step 2 Find the y-coordinate. It is found by substituting the x-coordinate into y = Ax² + Bx + C and then simplifying.

Note:

It can be shown that the y-coordinate of the vertex of a parabola is given by

Example 1

Find the vertex of the parabola: y = x² - 8x + 12

Solution

Here, A = 1, B = -8, and C = 12.

Step 1	Find the x-coordinate.	x
	Substitute 1 for A and -8 for B.	x
	Simplify.	x	= 4
Step 2	Find the y-coordinate. Substitute 4 for x. Simplify.	y y y y	= x2 - 8x + 12 = (4)2 - 8(4) + 12 = 16 - 32 + 12 = -4

Note:

When a parabola has two x-intercepts, the x-coordinate of the vertex always lies halfway between the x-intercepts.

Example 2

Find the vertex of the parabola: f(x) = -2x² - 12x - 18

Solution

Here, A = -2, B = -12, and C = -18.

Step 1	Find the x-coordinate.	x
	Substitute -2 for A and -12 for B.	x
	Simplify.	x	-3
Step 2	Find the y-coordinate. Substitute -3 for x. Simplify.	f(x) y y y	= -2x2 - 12x - 18 = -2(-3)2 - 12(-3) - 18 = -2(9) + 36 - 18 = 0

Note:

When a parabola has one x-intercept, the x-intercept is the x-coordinate of the vertex.