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Summary

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 $a<0$ ):

Maths; Graphs; KS4 Year 10; Quadratic graphs

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.

Learn with Basics

Length:

Unit 1

X and Y coordinates

Unit 2