New Zealand
Level 8 - NCEA Level 3

# Relation y^2 = x

Lesson

Let's consider the relation $y^2=x$y2=x.

Firstly, why are we calling this a relation, instead of a function?

Definition of a Function

A function is a relationship where each input value ($x$xvalue) has a single output value ($y$y value).

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 $x$x that we try. So since there are two outputs for every input, $y^2=x$y2=x can't be considered a function, just a relation.

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

When you look at the graph below, it should remind you of a far more familiar graph.

Does it remind you of a parabola (or quadratic function graph) on its side?

In fact, $y^2=x$y2=x is a reflection of $y=x^2$y=x2 over the line $y=x$y=x, and when this happens we call $y^2=x$y2=x the inverse function of $y=x^2$y=x2.

## How can we make $y^2=x$y2=x a function?

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

$y=\pm\sqrt{x}$y=±x

We can break this up into two separate functions: $y=\sqrt{x}$y=x and $y=-\sqrt{x}$y=x

If we graph each of these separately, we can see that each graph has one output for each input.

#### Worked Examples

##### Question 1

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

1. State the range of values of $x$x for which the relation is defined.

2. Complete the table for the values of $x$x.

 $x$x $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3
3. Plot the points from the table of values on the number plane.

4. Here is the graph of the curve that passes through the plotted points.

How many points on the graph correspond to any one particular value of $x$x, for $x>0$x>0?

##### Question 2

Consider the graph of the relation $x=-y^2$x=y2.

1. Which two of the following functions can be combined together to form the same graph as $x=-y^2$x=y2?

$y=-\sqrt{-x}$y=x

A

$y=-\sqrt{x}$y=x

B

$y=\sqrt{x}$y=x

C

$y=\sqrt{-x}$y=x

D

$y=-\sqrt{-x}$y=x

A

$y=-\sqrt{x}$y=x

B

$y=\sqrt{x}$y=x

C

$y=\sqrt{-x}$y=x

D
2. Use technology (or otherwise) to obtain graphs of the two functions $y=\sqrt{-x}$y=x and $y=-\sqrt{-x}$y=x, and then use these graphs to answer the following question:

Over which values of $x$x is the relation defined?

$x<0$x<0

A

$x\ge0$x0

B

$x>0$x>0

C

$x\le0$x0

D

$x<0$x<0

A

$x\ge0$x0

B

$x>0$x>0

C

$x\le0$x0

D

##### Question 3

The graph of $x=y^2$x=y2 has been plotted, along with points $A$A$\left(1,1\right)$(1,1) and $B$B$\left(4,-2\right)$(4,2). The point $F$F$\left(\frac{1}{4},0\right)$(14,0) is called the focus and the line $x=-\frac{1}{4}$x=14 is called the directrix.

1. Determine the distance between points $A$A and $F$F.

2. Determine the perpendicular distance between point $A$A and the directrix.

3. Determine the distance between points $B$B and $F$F.

4. Determine the perpendicular distance between point $B$B and the directrix.

5. Complete the gaps to make the statement true:

The graph of $\editable{}=\left(\editable{}\right)^2$=()2 represents the collection of points that are equidistant from the fixed point with coordinates $\left(\editable{},\editable{}\right)$(,), and the fixed line $\editable{}=\editable{}$=. The fixed point is called the focus and the fixed line is called the directrix.

### Outcomes

#### M8-1

Apply the geometry of conic sections

#### 91573

Apply the geometry of conic sections in solving problems