Question

How do you graph `y = x^2 + 3x + 2`?

Answer 1

The obvious method, but the least interesting is to choose `x` values, work out the corresponding `y` values and then plot the points. At least 5 or 7 points would be required to ensure some accuracy.

Explanation:

However, a more interesting method is to determine specific important points on the graph using algebra and then plot only those. The points are: the `y -` intercept, the two `x -` intercepts and the turning point (TP), which lies on the axis of symmetry.

For the `y`-intercept make `x` = 0.
This will give the constant as the `y`-intercept. `y` = 2.

For the `x`-intercepts make `y` = 0 and factorise.
The `x`-intercepts are also known as the roots of the equation.

If `y = (x+2)(x+1)`, then `0 = (x+2)(x+1)`
So `x =-2, or x = -1`

The line of symmetry lies halfway between the `x-`intercepts, giving `((-2)+( -1))/2 = -1.5`
This can also be found from the formula `x = -b/2a`

The `y` value of the TP can be calculated by substituting `x= 1.5` into the original equation, giving `y = -1/4`.

Plot these 4 points (0,2) ; (-2,0) ; (-1, 0) and (-1`1/2, -1/4`).
Join them with a smooth hand-drawn line and you have the graph.