Quadratic graphs

​​In a nutshell

Quadratic graphs have equations of the form $$y=ax^2+bx+c$$ where $$a$$, $$b$$ and $$c$$​ are constants. When plotted, they are curved like a $$\cup$$ or like a $$\cap$$. Plotting points on a quadratic curve is a case of inserting an $$x$$- or a $$y$$-coordinate of some point on the curve into the equation of the curve to find the corresponding $$y$$- or $$x$$-coordinate(s) of the point.

What do quadratic graphs look like?

Quadratic graphs have either a $$\cup$$-shape (a positive quadratic, when the $$a$$-term in the equation is positive) or a $$\cap$$-shape (a negative quadratic, when $$a<0$$​):

Plotting points on a quadratic graph

As with plotting points on a linear graph, to plot a point on a quadratic curve, insert a coordinate into the equation of the curve to find the corresponding point(s). Because of the shape of quadratic curves, it follows that for a single $$y$$-coordinate, there can be as many as two corresponding $$x$$-coordinates.

procedure

Example 1

Find the $$y$$-coordinate of the point on the quadratic curve $$y=x^2+5x+6$$ that has $$x$$-coordinate $$4$$.

Insert $$x=4$$ into the equation of the curve and solve:

$$y=x^2+5x+6$$​​

$$y=(4)^2+5(4)+6$$

$$y=16+20+6$$

$$y=\underline{42}$$

Thus the point $$(4,42)$$ is on the curve.

Example 2

Find all of the points on the curve $$y=x^2-4x-5$$​ that have $$y$$-coordinate $$7$$.

Insert $$y=7$$ into the equation of the curve and solve:

$$y=x^2-4x-5$$

$$7=x^2-4x-5$$

$$0=x^2-4x-12$$

$$0=(x-6)(x+2)$$

$$x=6$$ and $$x=-2$$

Thus the points $$\underline{(6,7)}$$ and $$\underline{(-2,7)}$$ are points on the curve.