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

1.

Insert the given coordinate into the equation of the curve.

2.

Solve the equation obtained. If solving for $y$, this will have one solution. If solving for $x$ this will have either one or two solutions. If you find there is no solution, then the given coordinate does not correspond to a point on the curve.

3.

The given coordinate and the obtained coordinate(s) give the coordinates of some point(s) on the curve.

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.

Quadratic graphs have equations of the form y=ax^2+bx+c where a, b and c are constants.

How do you find a point on a quadratic curve?

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).

What shapes do quadratic curves have?

Quadratic graphs have either a ∪-shape or a ∩-shape.