How to find the intercept form of a quadratic function?

To find the intercept form of a quadratic function, follow these steps:

Here's a detailed guide on how to convert a quadratic function to intercept form, also known as factored form. The intercept form of a quadratic equation is given by:

y = a(x – p)(x – q)

where p and q are the x-intercepts of the parabola.

Steps to Find the Intercept Form

Here's a step-by-step guide on how to find the intercept form of a quadratic function, based on the provided reference:

1. Find the x-intercepts: Determine the points where the parabola intersects the x-axis. These are also known as the roots or zeros of the quadratic function. You can find them by setting y = 0 and solving for x.
2. Find one other point that the graph passes through: Choose any point (x, y) on the parabola that is not one of the x-intercepts. This point will be used to find the value of a.
3. Substitute the x-intercepts for p and q in the intercept form y = a(x – p)(x – q): Replace p and q with the values of the x-intercepts you found in Step 1.
4. Substitute the point for x and y: Plug the coordinates (x, y) of the point you found in Step 2 into the equation from Step 3.
5. Solve for a: Simplify the equation and solve for a. This value determines the direction and stretch of the parabola.
6. Write the function by substituting p, q, and a into the intercept form: Substitute the values of a, p, and q into the intercept form y = a(x – p)(x – q). This is the intercept form of the quadratic function.

Example

Let's say you have a parabola with x-intercepts at x = 2 and x = -1, and it passes through the point (1, -4).

1. p = 2, q = -1
2. The point is (1, -4).
3. Substitute into the intercept form: y = a(x - 2)(x + 1)
4. Substitute the point (1, -4): -4 = a(1 - 2)(1 + 1)
5. Solve for a:
   -4 = a(-1)(2)
   -4 = -2a
   a = 2
6. Write the intercept form: y = 2(x - 2)(x + 1)

Summary

The intercept form is useful for quickly identifying the x-intercepts of a quadratic function, which can be helpful in graphing the parabola and solving related problems. Remember to find the x-intercepts, an additional point, and then solve for the leading coefficient a.