Graph in Vertex Form: Quadratic Equations

Graph in Intercept (Factored) Form: Quadratic Equations - How it Works - Video

General Graph Information

General Graph Information:

Here we have the vertex formula of a quadratic equation, y = a(x - h)² + k. The a-value is the same as the other forms: standard form and intercept form. In the vertex form, we are given the vertex, (h, k).

With the vertex, we are given two facts: the axis of symmetry and the maximum/minimum. The axis of symmetry is the h-value and maximum/minimum value is the k. If the a-value is positive, then k is a minimum. If the the a-value is negative, then k is a maximum.

The axis of symmetry is still the mirror line where each point to the left and right is the same distance.

Maximum vs Minimum

Maximum vs Minimum:

To find out with the quadratic equation has a maximum or minimum point, we can look at the a-value. If a is positive, then the y-values go up, so the graph has a minimum. If the a is negative, then the y-values go down, so the graph has a maximum.

To the find the maximum/minimum value we look at the y-part of the vertex.

Example 1

Example 1:

Here we have the equation, y = (x - 3)² - 1.

We can rewrite the equation as y = 1 * (x - 3)² - 1. Now we know the a-value, a = 1. Unfortunately, we can only determine the b and c values when the equation is in standard form. So we do not know them. If you want to find them, you can always expand the binomial.

Remember the equation is y = a * (x - h)² + k. So k = 3 and h = -1. This means our vertex is at (3, -1).

Now we know two facts, the axis of symmetry and the maximum/minimum. Since our a-value is positive, we have a minimum, -1.

Example 1 continued:

Our equations is y = (x - 3)² - 1. We already have the vertex. Now there are multiples to graph here. The way that we are going to do it use one directly to the left of the vertex and one point directly to the right of the vertex. Or if you prefer directly to the left and directly to the right of the axis of symmetry.

So our axis of symmetry is x = 3. So let's choose 2 and 4 as our inputs to find two points so we can graph the quadratic equation. Now let's find our outputs or the y-values.

Now we have two more points to graph this parabola. If you use the axis of symmetry then you only need to calculate one point and then find the distance to the axis of symmetry.

Now all we need to do is connect the dots.

Example 2

Example 2:

Here we have the equation, y = -(x + 2)² + 3.

We can rewrite the equation as y = -1 * (x + 2)² + 3. Now we know the a-value, a = -1. Unfortunately, we can only determine the b and c values when the equation is in standard form. So we do not know them. If you want to find them, you can always expand the binomial.

Remember the equation is y = a * (x - h)² + k. So k = -2 and h = 3. This means our vertex is at (-2, 3).

Now we know two facts, the axis of symmetry and the maximum/minimum. Since our a-value is negative, we have a maximum, 3.

Example 2 continued:

Our equations is y = -(x + 2)² + 3. We already have the vertex. Now there are multiples to graph here. The way that we are going to do it use one directly to the left of the vertex and one point directly to the right of the vertex. Or if you prefer directly to the left and directly to the right of the axis of symmetry.

So our axis of symmetry is x = -2. So let's choose -3 and -1 as our inputs to find two points so we can graph the quadratic equation. Now let's find our outputs or the y-values.

Now we have two more points to graph this parabola. If you use the axis of symmetry then you only need to calculate one point and then find the distance to the axis of symmetry.

Now all we need to do is connect the dots.

Live Worksheet

Teacher - Edpuzzle Link