Plotting Quadratic Equations

These topics are covered in the Mathematics syllabus in preparation for the University of Malta MATSEC O level exam. However a strong knowledge of such topics will prove to be extremely important and useful for the workings in the A level topics. These topics are intended as a revision and brush-up for students starting their A level Pure Mathematics in Malta.

Quadratic equations have the following form:

y=ax^2+bx+c

where $a,b$ and $c$ are some numbers. However $a$ must not be zero ( $a\neq 0$ ). In fact, if $a$ where $0$ , the equation above becomes $y=bx+c$ which is the equation of a straight line and not a quadratic. The following are examples of quadratic equations:

\begin{aligned}(1)& \ y=x^2+3x-1\\(2)&\ y=3x^2+1\\(3)& \ y=-x^2+x\\(4)& \ y=(x-1)^2\end{aligned}

In quadratic (1), we have $a=1$ , $b=3$ and $c=-1$ . In quadratic (2), we note that the term in $x$ is missing (i.e. $b=0$ ). Still this is a quadratic equation because the term in $x^2$ is present. Similarly in quadratic (3), the constant term is missing (i.e. $c=0$ ). Still this remains a quadratic equation because the term in $x^2$ is present. Finally, quadratic (4) is a quadratic because when we expand (open up) the brackets we obtain $y=x^2-2x+1$ and this has the term in $x^2$ present.

The two different shapes of the quadratic equation

There are two different shapes that a quadratic equation can take, when plotted. The first shape is the U-shape where the quadratic has the shape of the letter “U” and the second shape is the N-shape where the quadratic has the shape of the letter “n”. The shape of the quadratic equation $y=ax^2+bx+c$ is determine only by the coefficient of $x^2$ . If $a>0$ , the quadratic is U-shaped. If $a<0$ , the quadratic is N-shaped. The following diagram helps you remember this rule.

Example: Determine the shape of the following quadratics:

(a) $y=x^2+3x-1$

We have $a=1$ , thus $a>0$ . So the quadratic is U-shaped. In fact the plot of this quadratic is the following:

(b) $y=-2x^2+4$

We have $a=-2$ , thus $a<0$ . So the quadratic is N-shaped. In fact the plot of this quadratic is the following:

Let us expand this quadratic equation:

\begin{aligned}y&=(x-1)^2\\y&=(x-1)(x-1)\\y&=x^2-2x+1\\ & \end{aligned}

We have $a=1$ , thus $a>0$ . So the quadratic is U-shaped. In fact the plot of this quadratic is the following:

(b) $y=-(x+1)^2$

Let us expand this quadratic equation:

\begin{aligned}y&=-(x+1)^2\\y&=-(x+1)(x+1)\\y&=-(x^2+2x+1)\\y&=-x^2-2x-1\end{aligned}

We have $a=-1$ , thus $a<0$ . So the quadratic is N-shaped. In fact the plot of this quadratic is the following:

Plotting Quadratic Equations

We will plot quadratic equations by using just 3 steps as follows:

Step 1: We find the $x$ -intercept(s) (also known as the root(s)). This is done by letting $y=0$ in the equation, factorising the quadratic and solving for $x$ .

Step 2: We find the $y$ -intercept (the point where the graph crosses the $y$ -axis). This is done by letting $x=0$ in the equation and solving for $y$ .

Step 3: We check whether the curve is U-shaped or N-shaped and draw a curved line through the $x$ -intercept(s) and the $y$ -intercept. Note that the graph reaches its minimum/maximum point exactly between the roots.

A note regarding the $x$ -intercept(s): We will have either two different $x$ -intercepts, or just one distinct $x$ -intercept or else no $x$ -intercepts. The case where we have no $x$ -intercepts will be discussed in more details in the A-level topic on Differentiation.

Example 1: Plot the following equation: $y=x^2+2x-3$

In Step 1, let $y=0$ ,

\begin{aligned}0&=x^2+2x-3\\0&=(x+3)(x-1) \end{aligned}

\begin{aligned} x+3&= 0  &  x-1& = 0\\x&=-3 & x&=1\end{aligned}

Therefore the roots are $(-3,0)$ and $(1,0)$

In Step 2, we let $x=0$ ,

\begin{aligned}y&=0^2+2(0)-3\\y&=-3 \end{aligned}

Therefore the $y$ -intercept is $(0,-3)$ . We draw a curve through the roots and the $y$ -intercept that is symmetric between the roots. Note that since $a>0$ , the curve will be U-shaped.

Example 2: Plot the following equation: $y=-2x^2+4$

In Step 1, let $y=0$ ,

\begin{aligned}0&=-2x^2+4\\0&=x^2-2 \text{ (divide both sides by }-2 )\\0&=(x+\sqrt{2})(x-\sqrt{2})\end{aligned}

\begin{aligned} x+\sqrt{2}&= 0  &  x-\sqrt{2}& = 0\\x&=-\sqrt{2} & x&=\sqrt{2}\end{aligned}

Therefore the roots are $(-\sqrt{2},0)$ and $(\sqrt{2},0)$

In Step 2, we let $x=0$ ,

\begin{aligned}y&=-2(0^2)+4\\y&=4 \end{aligned}

Therefore the $y$ -intercept is $(0,4)$ . We draw a curve through the roots and the $y$ -intercept that is symmetric between the roots. Note that in this example, $a=-2>0$ , hence the curve will be N-shaped.

Example 3: Plot the following equation: $y=(x-1)^2$

In Step 1, let $y=0$ ,

\begin{aligned}0&=(x-1)^2\\0&=(x-1)(x-1)\end{aligned}

\begin{aligned} x-1&= 0  &  x-1& = 0\\x&=1 & x&=1\end{aligned}

Note that here we have just one distinct root (in other words, a repeated root) which is given by the coordinates $(1,0)$ .

In Step 2, we let $x=0$ ,

\begin{aligned}y&=(0-1)^2\\y&= (-1)^2\\y&=1\end{aligned}

Therefore the $y$ -intercept is $(0,1)$ .

For Step 3, let us expand the quadratic in order to determine its shape.

\begin{aligned}y&=(x-1)^2\\y&=(x-1)(x-1)\\y&=x^2-2x+1\end{aligned}

Since $a=1>0$ , the curve will be U-shaped. We draw a U-shaped curve passing through the root and the $y$ -intercept.

Summary

We have learnt how to plot a quadratic equation in 3 simple steps. In Step 1, we find the $x$ -intercepts (roots), in Step 2, we find the $y$ -intercept and finally in Step 3, we determine the shape of the quadratic (U-shape or N-shape) and plot the curve.

Note that in the case of a quadratic equation that has no roots (no $x$ -intercepts), we need to find the minimum or maximum point in order to be able to get a decent plot. However this case will be covered in the topic of Differentiation.

Exercises

Plot the following quadratic equations:

a) $y=x^2-x$

b) $y=4-x^2$

c) $y=x^2$

d) $y=x^2+4x+4$

e) $y=2x^2-2x+\frac{1}{2}$

f) $y=2(x+1)(x+4)$

Plotting Quadratic Equations

The two different shapes of the quadratic equation

Example: Determine the shape of the following quadratics:

Plotting Quadratic Equations

Example 1: Plot the following equation: y=x2+2x−3y=x^2+2x-3y=x2+2x−3

Example 2: Plot the following equation: y=−2x2+4y=-2x^2+4y=−2x2+4

Example 3: Plot the following equation: y=(x−1)2y=(x-1)^2y=(x−1)2

Summary

Exercises

Example 1: Plot the following equation: $y=x^2+2x-3$

Example 2: Plot the following equation: $y=-2x^2+4$

Example 3: Plot the following equation: $y=(x-1)^2$