Graph in Intercept (Factored)  Form: Quadratic Equations

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

General Graph Information

General Graph Information:

Here we have the standard formula of a quadratic equation, y = a(x - p)(x - q). The a value is the same as the other forms: standard form and vertex form. Before we do any math, we know two points, (p, 0) and (q, 0), which are the x-intercepts.

In intercept form, the next bit of information we can find is the axis of symmetry. In this case, with the parent function, the axis of symmetry is x = 0. The great thing about the axis of symmetry is that it tells us the x-part of the vertex. And, to find the y-part we substitute the value of the axis of symmetry into the equation to find the y-part, which is the maximum of minimum. 

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 + 1)(x - 3).

The first step rewrite it as y = 1 * (x + 1)(x - 3). Now we know the a-value, which is a = 1. This helps us determine if the graph does up or down. Since it is positive we know it goes up. Or the y-values continue to increase. If the quadratic is in this form, we don't know the b or c values. You can distribute the two binomials and find the b and c values if you want.

Now we can separate each binomial and set it equal to 0 to find the x-intercepts. So we have (x + 1) = 0 and (x - 3) = 0. With (x + 1), we drop the parentheses and subtract 1 on both sides to get x = -1. With (x - 3), we drop the parentheses and add 3 on both sides to get x = 3. Now, we have the x-intercepts: (-1, 0) and (3, 0). Let's plot them on the graph.

Remember the axis of symmetry is where we fold the graph in half so each point on the same horizontal line is halfway to the axis of symmetry. We can either count the spaces and find the middle. Or we can find the average of the two numbers, which is 1, because -1 + 3 is 2 and 2 divided by 2 is 1. So the axis of symmetry is x = 1.

Now we need to find the vertex. Is it below or above the points we already found? It is below since our a-value is positive, so we have a minimum.

Example 1 continued:

Now that we the axis of symmetry, x = 1. We can find the vertex, since the axis of symmetry is the x-part of the vertex. So to find the y-part of the vertex we substitute that value in the equation.

Now we can combine the x and y for our vertex: (1, -4). We have our points. Now we can plot them and connect the dots. The last fact we are going to talk about is the minimum value, which is -4, because that is the y-part of the vertex. The vertex in this case is the lowest point of the graph, so there is not any value below it.

Example 2

Example 2:

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

The first step rewrite it as y = -2 * (x + 3)(x - 1). Now we know the a-value, which is a = 1. This helps us determine if the graph does up or down. Since it is negative we know it goes down. Or the y-values continue to decrease. If the quadratic is in this form, we don't know the b or c values. You can distribute the two binomials and find the b and c values if you want.

Now we can separate each binomial and set it equal to 0 to find the x-intercepts. So we have (x + 3) = 0 and (x - 1) = 0. With (x + 3), we drop the parentheses and subtract 3 on both sides to get x = -3. With (x - 1), we drop the parentheses and add 1 on both sides to get x = 1. Now, we have the x-intercepts: (-3, 0) and (1, 0). Let's plot them on the graph.

Remember the axis of symmetry is where we fold the graph in half so each point on the same horizontal line is halfway to the axis of symmetry. We can either count the spaces and find the middle. Or we can find the average of the two numbers, which is 1, because -3 + 1 is -2 and -2 divided by 2 is -1. So the axis of symmetry is x = -1.

Now we need to find the vertex. Is it below or above the points we already found? It is below since our a-value is negative, so we have a maximum.

Example 2 continued:

Now that we the axis of symmetry, x = -1. We can find the vertex, since the axis of symmetry is the x-part of the vertex. So to find the y-part of the vertex we substitute that value in the equation.

Now we can combine the x and y for our vertex: (-1, -8). We have our points. Now we can plot them and connect the dots. The last fact we are going to talk about is the maximum value, which is -8, because that is the y-part of the vertex. The vertex in this case is the highest point of the graph, so there is not any value above it.

Live Worksheet

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