4.05 Parabolas

The word parabola is usually associated with a function we're very familiar with: the quadratic function.

 

Quadratic function recap

Quadratic functions can be written in each of the following forms:

• $y=ax^2+bx+c$y=ax2+bx+c, which is known as the general form
• $y=a\left(x+\alpha\right)\left(x+\beta\right)$y=a(x+α)(x+β), which is known as the factored form
• $y=a\left(x+h\right)^2+k$y=a(x+h)2+k, which is known as the turning point form

Next, take a look at what happens if the variables $x$x and $y$y are swapped in these equations.

 

The graph of the relation $y^2=x$y2=x is also a parabola

Consider the relation $y^2=x$y2=x.

Firstly, why is this called a relation, rather than a function? Recall the following:

Let's look at a few points on the graph:

• If $y^2=1$y2=1, then $y$y can take on the values of $-1$−1 and $1$1. This means that for $x=1$x=1 there are two $y$y value outputs, rather than just one.
• If $y^2=4$y2=4, then y can take on the values of $-2$−2 and $2$2. This means that for $x=4$x=4 there are two $y$y value outputs, rather than just one.

In fact, this will be true for all values of $x>0$x>0. Since there are two outputs for every input, $y^2=x$y2=x is not a function.

 

The graph of $y^2=x$y2=x

When you look at the graph below, it should remind you of another, familiar graph.

This looks like the graph of a quadratic function that has been rotated to lie on its side.

In fact, $y^2=x$y2=x is a reflection of $y=x^2$y=x2 across the line $y=x$y=x.

 

Restricting $y^2=x$y2=x to make a function

Let's rearrange $y^2=x$y2=x so that $y$y is the subject:

$y=\pm\sqrt{x}$y=±√x

This can be broken up into two separate functions: $y=\sqrt{x}$y=√x and $y=-\sqrt{x}$y=−√x

Sketching the graphs of each of these separately, notice that each graph now has one output for each input.

The graph of $y=\sqrt{x}$y=√x

The graph of $y=-\sqrt{x}$y=−√x

 

Practice question

Question 1

 

Translation of parabolas

Translations can be applied to the graph of $x=y^2$x=y2, just as for quadratic functions, and form an equation analogous to turning point form: $x=a\left(y-k\right)^2+h$x=a(y−k)2+h

Let's look at the orientation possibilities of parabolas that can be constructed using the forms:

$y-k=a\left(x-h\right)^2$y−k=a(x−h)2 and $x-h=a\left(y-k\right)^2$x−h=a(y−k)2

Examples of these are shown below. Look carefully at the coordinates of each vertex and how that matches up with the corresponding equation.

For instance, in diagram (B), there is a horizontal translation of $2$2 units to the left and $5$5 units up. There is also a reflection to deal with - the parabola is opening downward, and the function becomes more negative as the values of $x$x move away from the line of symmetry $x=-2$x=−2. This matches the form of the corresponding equation, $y-5=-\left(x+2\right)^2$y−5=−(x+2)2.

Summary:

• Each graph has its vertex at $\left(h,k\right)$(h,k)
• If the graph is of the form $y=...$y=..., then the axis of symmetry is vertical with equation $x=h$x=h
• If the graph is of the form $x=...$x=..., then the axis of symmetry is horizontal with equation $y=k$y=k
• If $a>0$a>0 the graph opens upwards or to the right
• If $a<0$a<0 the graph opens downward or to the left

So, to construct an equation of a parabola that has its vertex at $\left(-3,4\right)$(−3,4), and opening to the left, we could choose $x+3=-\left(y-4\right)^2$x+3=−(y−4)2. This is not the only possible choice - any equation of the form $x+3=-a\left(y-4\right)^2$x+3=−a(y−4)2 (where $a$a is a positive number) will also have those attributes. The coefficient $a$a is a dilation factor and will dilate the graph by a factor of $a$a from the $y$y-axis.

 

Different forms

Just as with quadratic functions, there may be a need to graph parabolas from general form: $y=ax^2+bx+c$y=ax2+bx+c or $x=ay^2+by+c$x=ay2+by+c. Recall that the general form can be converted to turning point form by completing the square. Turning point form can also be found from the general form by first finding the equation of the line of symmetry, using the formula $x=\frac{-b}{2a}$x=−b2a​ for vertical parabolas or $y=\frac{-b}{2a}$y=−b2a​ for horizontal parabolas. Then, substitute this back into the equation to find the turning point.

 

Practice questions

Question 2

Question 3

Question 4