Finding the Vertex, Minimum, Maximum, and Axis of Symmetry in a Quadratic Function

Math can feel like a puzzle sometimes, but once you break it down, it’s not so bad. Let’s talk about some key parts of a quadratic function: the vertex, minimum or maximum, and the axis of symmetry. If you’re dealing with a parabola (that U-shaped curve), knowing these will help you understand how it moves and where it’s going.

What’s a Vertex?

The vertex is the highest or lowest point of a parabola. It’s like the peak of a mountain or the bottom of a valley. You find it by using the formula:

x = -b / 2a

Once you get that x-value, plug it into the function to find the y-value. That gives you the vertex (x, y).

Example: Say you have y = 2x² - 4x + 3.

• Find x: x = -(-4) / (2*2) = 4 / 4 = 1

• Find y: y = 2(1)² - 4(1) + 3 = 2 - 4 + 3 = 1

• So, the vertex is (1,1).

Minimum or Maximum?

Look at the a value (the number in front of x²):

• If a is positive, the parabola opens up (like a U), so the vertex is a minimum.

• If a is negative, the parabola opens down (like an upside-down U), so the vertex is a maximum.

Using our example, a = 2, which is positive. So, (1,1) is a minimum.

Axis of Symmetry? What’s That?

The axis of symmetry is an invisible vertical line that cuts the parabola right down the middle. The equation for this line is x = the x-value of the vertex.

For our example, the vertex was (1,1), so the axis of symmetry is x = 1. That means the left and right sides of the parabola mirr

Another example: a maximum

Say you have y = negative x squared plus 4x plus 1. Here a is negative 1 and b is 4.

• Find x: x = negative 4 divided by (2 times negative 1) = negative 4 divided by negative 2 = 2

• Find y: y = negative (2 squared) plus 4(2) plus 1 = negative 4 plus 8 plus 1 = 5

• So the vertex is (2, 5).

Since a is negative, the parabola opens down. So (2, 5) is a maximum. The axis of symmetry is x = 2.

Mistakes to watch for

• Watch the sign of b. In the first example, b was negative 4, so negative b became positive 4.

• Do not forget to plug the x value back in to get y. The vertex has two numbers.

• Do not mix up the vertex and the zeros. The zeros are where the graph crosses the x axis.

Practice problems

1. y = x squared minus 6x plus 5. Answer: vertex (3, negative 4), a minimum.

2. y = negative 2x squared plus 8x. Answer: vertex (2, 8), a maximum.