2.01 Characteristics of parabolas

Equations such as $y=x^2$y=x2 $y=5-x^2$y=5−x2, $y=x^2-3x-4$y=x2−3x−4 and $y=x\left(x-1\right)$y=x(x−1) are called quadratics, and they all result in a symmetric curve called a parabola.

In general, a quadratic function has the form $y=ax^2+bx+c$y=ax2+bx+c, where $b$b and $c$c can be any number and $a$a can be any number except for zero.

Graph of $y=x^2$y=x2

The basic quadratic function is $y=x^2$y=x2, and using this equation we can create a table of values:

$x$x	$-3$−3	$-2$−2	$-1$−1	$0$0	$1$1	$2$2	$3$3
$y$y	$9$9	$4$4	$1$1	$0$0	$1$1	$4$4	$9$9

 

Graphing the parabola, we get:

 

All functions of the form $y=ax^2+bx+c$y=ax2+bx+c result in parabolas that share some common features.

Symmetry

All parabolas are symmetric about a line called the axis of symmetry. For the parabola $y=x^2$y=x2, the axis of symmetry is the $y$y-axis. Another way we can describe this is to say the axis of symmetry is the line with equation $x=0$x=0.

Turning point

The point on a parabola where the curve turns and changes from decreasing to increasing (or vice‑versa) is called the turning point. It is the point where the curve reaches either the minimum or maximum value of $y$y.

• The graph of $y=x^2$y=x2 has a turning point at $\left(0,0\right)$(0,0), where the $y$y-value is a minimum.
• The graph of $y=-x^2$y=−x2 also has a turning point at $\left(0,0\right)$(0,0), where the $y$y-value is a maximum.

Concavity

Concavity refers to the general shape of the graph.

Notice that the graph of $y=x^2$y=x2 opens upwards, whereas the graph of $y=-x^2$y=−x2 opens downwards.

For an equation of the form $y=ax^2+bx+c$y=ax2+bx+c:

• If $a$a is positive, the parabola opens upwards. We say the parabola has positive concavity or is concave up.
• If $a$a is negative, the parabola opens upwards. We say the parabola has negative concavity or is concave down.

So the graph of $y=x^2$y=x2 is concave up (opens upwards) and the graph of $y=-x^2$y=−x2 is concave down (opens downwards).

Intercepts

• $y$y-intercept:

This occurs when $x=0$x=0. All parabolas have a $y$y-intercept. That is, they eventually cross the $y$y-axis.For parabolas with an equation in the form of $y=ax^2+bx+c$y=ax2+bx+c, the constant $c$c represents this $y$y-intercept.

• $x$x-intercept(s):

These occur when $y=0$y=0. Not all parabolas have an $x$x-intercept, as shown in the graph below. Some parabolas cross the $x$x-axis twice ($C$C), some touch the x-axis once ($A$A) and some miss it completely ($B$B). The number of $x$x-intercepts depends on the equation of the parabola.

Here are three parabolas where each has a different number of $x$x-intercepts.

Parabolas can have either zero, one or two $x$x-intercepts.

The equation of a parabola

Considering the general equation of a quadratic, $y=ax^2+bx+c$y=ax2+bx+c, notice that:

• They all have a term with $x^2$x2.
• They can have a term with $x$x, but don't have to.
• They can have a constant term (a term without any variables), but don't have to.
• They cannot have any terms that contain negative or fractional powers.

This means that the highest power of the function is $2$2.

For example, the equation $y=2x^2-5$y=2x2−5 is a quadratic and it graph is a parabola, but $y=2x^2-x^{-1}-5$y=2x2−x−1−5 and $y=x^3-x^2+x-1$y=x3−x2+x−1 are not quadratics.

 

Let's now look at some parabolas and their features.

Graphs of $y=ax^2$y=ax2

This graph shows three parabolas corresponding to the equations $y=x^2$y=x2, $y=3x^2$y=3x2 and $y=-2x^2$y=−2x2:

These parabolas share certain features:

• They are all symmetric about the $y$y-axis.
• They all have a turning point at the origin.

There are some differences as well:

• The quadratic $y=-2x^2$y=−2x2 has a parabola with a maximum turning point, while the other two have minimum turning points.
• The quadratic $y=3x^2$y=3x2 has a narrower parabola than the quadratic $y=x^2$y=x2.

 

Graphs of $y=x^2+c$y=x2+c

This graph shows three parabolas corresponding to the equations $y=x^2$y=x2, $y=x^2+2$y=x2+2 and $y=x^2-4$y=x2−4:

These parabolas share certain features:

• They are all symmetric about the $y$y-axis.
• They all have a turning point, which lies on the axis of symmetry (the $y$y-axis). These points are also the $y$y-intercepts of the parabolas.

There are some differences as well:

• Their turning points (and therefore their $y$y-intercepts) are at different locations.
• They each cross the $x$x-axis a different number of times.

 

Graphs of $y=ax^2+bx+c$y=ax2+bx+c

This graph shows two more parabolas, which correspond to the equations $y=2x^2+4x$y=2x2+4x and $y=x^2-2x+3$y=x2−2x+3:

Can you identify the axis of symmetry and the turning point for each parabola?

 

 

Practice questions

Question 1

QUESTION 2

QUESTION 3