3. Algebraic Relations & Functions

Lesson

Recall the coordinate plane is made up of a horizontal and a vertical axis and has $4$4 sections called quadrants. We name the quadrants starting in the top right whit quadrant 1 and continuing in the counter-clockwise direction as shown below.

Remember!

- The horizontal axis is called the$x$
`x`-axis and runs from left the right. - The vertical axis is called the $y$
`y`-axis and runs up and down. - The two lines meet at the origin, which has coordinates $\left(0,0\right)$(0,0).

A pair of coordinates describes a point's position away from the origin. A negative coordinate indicates a direction of left or down from the origin. A positive coordinate indicates a direction of right or up from the origin.

When we plot a pair of coordinates, we draw them on a coordinate plane. For example, to plot the point $\left(4,-2\right)$(4,−2), we would start at the origin, move $4$4 spaces to the right and $2$2 spaces down before plotting the point.

To name coordinates, we write the horizontal value, then the vertical value that a point is away from the origin. Remember, points are written as ordered pairs $\left(x,y\right)$(`x`,`y`). We can always remember this because $x$`x` comes before $y$`y` in the alphabet, so we should do the same in our ordered pair!

Use the applet below to practice plotting points. You will get a message when you have placed a point in the correct spot. Press the refresh button in the top right corner to get a new set of points to plot.

Consider the points $A$`A` and $B$`B`.

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Give the coordinates of the plotted points.

$A\left(\editable{},\editable{}\right)$ `A`(,)$B\left(\editable{},\editable{}\right)$ `B`(,)Which axis do points $A$

`A`and $B$`B`lie on?The $x$

`x`-axisAThe $y$

`y`-axisBThe $x$

`x`-axisAThe $y$

`y`-axisB

Consider the point $\left(6,-8\right)$(6,−8).

Plot the point on the coordinate plane.

Loading Graph...In which quadrant does the point $\left(6,-8\right)$(6,−8) lie?

2nd quadrant

A4th quadrant

B1st quadrant

C3rd quadrant

D2nd quadrant

A4th quadrant

B1st quadrant

C3rd quadrant

D

A relation is a set of ordered pairs which represent a relationship. For example, if we think of the names of the people in a math class and their heights this information represents a relation. If we chose a specific height (like $162$162 cm), we could list all the names of the people who are this tall and there may be more than one person. Let's say someone came to the class looking for the person who was $162$162 cm tall, that description might fit four people! There's not one clear answer. This data could be expressed as a relation.

We can express the same relation in several different ways -- as a mapping, a set of ordered pairs, an input-output table, a graph in the coordinate plane, or as an equation in terms of x and/or y that describes a graph .

A mapping diagram shows how the input values are assigned one or more output values. Consider the mapping below:

We can write an input-output table from the mapping, making sure that each pair is represented. A table can be laid out horizontally (like the one shown below) or vertically.

$x$x |
$-1$−1 | $0$0 | $1$1 | $2$2 |
---|---|---|---|---|

$y$y |
$2$2 | $0$0 | $2$2 | $4$4 |

This also corresponds to the set of ordered pairs $\left\{\left(-1,2\right),\left(0,0\right),\left(1,2\right),\left(2,4\right)\right\}${(−1,2),(0,0),(1,2),(2,4)}, which can be graphed in the coordinate plane, as shown below.

Express the relation $\left\{\left(2,2\right),\left(4,4\right),\left(6,3\right),\left(7,5\right)\right\}${(2,2),(4,4),(6,3),(7,5)} in the table below.

$x$ `x`$y$ `y`$2$2 $\editable{}$ $4$4 $\editable{}$ $6$6 $\editable{}$ $7$7 $\editable{}$

Consider the relation below:

$\left\{\left(-6,2\right),\left(4,1\right),\left(7,1\right),\left(7,8\right),\left(9,-7\right),\left(10,-10\right)\right\}${(−6,2),(4,1),(7,1),(7,8),(9,−7),(10,−10)}

Represent the relation on the coordinate plane.

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A relation is defined as follows.

$y=-4$`y`=−4 if $x$`x` is positive and $y=4$`y`=4 if $x$`x` is $0$0 or negative.

Complete the table.

$x$ `x`$-4$−4 $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $4$4 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points on the number plane.

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