Graphing quadratics in Intercept Form

Instructions

The intercept form of a quadratic equation is y = a(x - p)(x - q), where 'p' and 'q' are the x-intercepts and 'a' is the value that determines the vertical stretch and whether the parabola is upside down.

Now since you already have the x-intercepts of the parabola, you already know two of its points. The other point which you must know is the vertex.

In order to get the the vertex, follow the method described below:

  • Let the vertex be (h, k)
  • h = (p + q)/2
  • k = f(h)
In the above method, 'h' is the x coordinate of the vertex and 'k' is the y coordinate of the vertex. In order to get the value of 'h', use the formula `h = (p + q)/2` and in order to get the value of 'k', substitute the value of h in place of x in the quadratic equation.

After getting the vertex, now you have three points on the parabola, its vertex and two x-intercepts. Plot these points on a coordinate plane and join them with the help of a free hand curve. In order to make the curve more accurate or larger, you can get more points on the parabola by direct substition method (that is making a T chart)

Example

Graph the parabola y = 4(x - 1)(x - 3)

On comparing with y = a(x - p)(x - q), 
  • a = 4
  • p = 1
  • q = 3
So the x-intercepts are (1, 0) and (3, 0).
Let the vertex be (h, k), then 
`h = (p + q)/2 = (1 + 3)/2 = 2`
`k = 4(h - 1)(h - 3) = 4(2 - 1)(2 - 3) = 4(1)(-1) = -4`
The vertex is (2, 4)

Now plot the two x-intercepts and vertex on a coordinate plane and join them with the help of a free hand curve.

Note that you can get more points by direct substitution method in order to make the curve more accurate. This can be done by taking x values and plugging them in the equation as follows:

`x`
`y = 4(x - 1)(x - 3)`
0
12
0.5
a
1.5
-3
2.5
-3
4
12
Note that you can take any value of x in the above table. These are only some example values.

Graph `y = 4(x - 1)(x - 3)`

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