Graphing quadratics in Intercept Form - Solved Examples

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

Graph of `y = (x - 1)(x - 4)`
Comparing the equation with intercept form `y = a(x - p)(x - q)`, p = 1 and q = 4. So the x-intercepts are `(1, 0)` and `(4, 0)`.

Let the vertex be (h, k), then `h = (p + q)/2 = (1 + 4)/2 = 5/2`,

`k = (5/2 - 1)(5/2 - 4)`
`k = (3/2)(-3/2)`
`k = -9/4`

Therefore the vertex is `(5/2, -9/4)`. Now plot the vertex and x-intercepts and join them with a free hand curve

Graph `y = 3(x - 6)(x - 3)`


Graph of `y = 3(x - 6)(x - 3)`
Comparing the equation with intercept form `y = a(x - p)(x - q)`, p = 6 and q = 3. So the x-intercepts are `(6, 0)` and `(3, 0)`.

Let the vertex be (h, k), then `h = (p + q)/2 = (6 + 3)/2 = 9/2`,

`k = 3(9/2 - 6)(9/2 - 3)`
`k = 3(-3/2)(3/2)`
`k = -27/4`

Therefore the vertex is `(9/2, -27/4)`. Now plot the vertex and x-intercepts and join them with a free hand curve

Graph `y = -2(x - 4)(x - 5)`

Comparing the equation with intercept form `y = a(x - p)(x - q)`, p = 4 and q = 5. So the x-intercepts are `(4, 0)` and `(5, 0)`.
Graph `y = -2(x - 4)(x - 5)`

Let the vertex be (h, k), then `h = (p + q)/2 = (4 + 5)/2 = 9/2`,

`k = -2(9/2 - 4)(9/2 - 5)`
`k = -2(1/2)(-1/2)`
`k = 1/2`

Therefore the vertex is `(9/2, 1/2)`. Get some more points on the parabola by making table of x-y values

x
Y = -2(x - 4)(x - 5)
2
-12
3
-4
6
-4
7
-12

Now plot the vertex and x-intercepts and join them with a free hand curve

Graph `y = (x + 2)(x + 3)`

Comparing the equation with intercept form `y = a(x - p)(x - q)`, p = -2 and q = -3. So the x-intercepts are `(-2, 0)` and `(-3, 0)`.

Let the vertex be (h, k), then `h = (p + q)/2 = (-2 + -3)/2 = -5/2`,

`k = (-5/2 + 2)(-5/2 + 3)`
`k = (-1/2)(1/2)`
`k = -1/4`

Therefore the vertex is `(-5/2, -1/4)`. Now plot the vertex and x-intercepts and join them with a free hand curve.  Get some more points on the parabola by making table of x-y values

x
y = (x + 2)(x + 3)
-1
2
-4
2
0
6
-5
6
Now plot the vertex and x-intercepts and join them with a free hand curve
Graph of `y = (x + 2)(x + 3)`

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