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# The Vertex of a Parabola

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

 â€¢ If the parabola opens up, the vertex is the lowest point on the graph. This is called the minimum. â€¢ If the parabola opens down, the vertex is the highest point on the graph. This is called the maximum.
A vertical line drawn through the vertex is called the axis of symmetry.

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

We can use the following procedure to find the vertex of a parabola.

Procedure â€” To Find the Vertex of a Parabola y = Ax2 + Bx + C

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. yy y y = x2 - 8x + 12= (4)2 - 8(4) + 12 = 16 - 32 + 12 = -4
So, the vertex of y = x2 - 8x + 12 is (4, -4).

Note:

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

Here the x-intercepts are x =  2 and x = 6. The x-coordinate of the vertex, 4, is halfway between 2 and 6.

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
So, the vertex of = -2x2 - 12x - 18  is (-3, 0).

Note:

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

Here the x-intercept is x = -3 and the x-coordinate of the vertex is x = -3.