Plotting points, reflections and shapes

How to plot a point

To plot $(x, y)$:

1. Start at the origin.
2. Move $x$ units along the x-axis: to the right if $x$ is positive, to the left if it is negative.
3. From there, move $y$ units up if $y$ is positive, or down if it is negative.
4. Mark the point and write its name.

To plot $(-4, 3)$, go $4$ units left, then $3$ units up. You are in quadrant II, as the signs said.

When you plot several points, it is useful to use graph paper, and to choose a scale such as one square for one unit, or one square for ten units. Label the axes.

Reading coordinates from a graph

To read a point, do the reverse. Look straight down (or up) to the x-axis: that number is $x$. Look straight across to the y-axis: that number is $y$.

Plotting a picture

Plot the points $(1, 1)$, $(5, 1)$, $(5, 4)$ and $(1, 4)$, and join them in order. You get a rectangle. Now join $(1, 4)$, $(3, 6)$ and $(5, 4)$, and you have a roof on it: a house. Many pictures on a computer screen are made in the same way, from the coordinates of the corners.

Reflections

A reflection is the mirror image of a point.

Think of it this way. Reflecting in the x-axis flips the point up or down, so only $y$ changes sign. Reflecting in the y-axis flips it left or right, so only $x$ changes sign. Reflecting in the origin does both.

A reflection keeps the distance from the mirror. The point $(3, 2)$ is $2$ units above the x-axis, and its mirror image $(3, -2)$ is $2$ units below it.

Move the blue point into another quadrant. Its three images always fill the other three quadrants, one each.

Lengths along a line parallel to an axis

If two points have the same $y$ (they are on a horizontal line), the distance between them is the difference of the $x$ values, taken as a positive number: $|x_2 - x_1|$. For $(-3, 2)$ and $(5, 2)$ it is $|5 - (-3)| = 8$ units.

If two points have the same $x$ (a vertical line), the distance is $|y_2 - y_1|$.

Shapes from their corners

Three corners of a rectangle are $(1, 1)$, $(5, 1)$ and $(5, 4)$. The sides of a rectangle are parallel to the axes here, so the fourth corner has the $x$ of the first and the $y$ of the third: $(1, 4)$. The length is $4$ units and the width is $3$ units, so the area is $12$ square units.

For a parallelogram $ABCD$, the fourth corner is $D = A + C - B$, corner by corner. This is because the diagonals of a parallelogram cut each other in half, so $A + C = B + D$.

Example. Three corners of a parallelogram $ABCD$ are $A(1, 1)$, $B(5, 2)$ and $C(6, 5)$. Then $D = (1 + 6 - 5,\ 1 + 5 - 2) = (2, 4)$.

Check. The step from $A$ to $B$ is $(4, 1)$. The step from $D$ to $C$ is $(6 - 2, 5 - 4) = (4, 1)$. The same step, so $AB$ and $DC$ are parallel and equal. ✓

Practise with new numbers

Try it in Python

This program reflects a point in the x-axis, the y-axis and the origin.

Common mistakes

• Changing the wrong sign in a reflection. In the x-axis, only $y$ changes.
• Counting units wrongly across the origin. From $-3$ to $5$ is $8$ units, not $2$.
• Plotting $y$ before $x$. Move across first.
• Forgetting the scale. If one square is ten units, a point at $(30, 20)$ is three squares across.

Remember these

• To plot $(x, y)$: move $x$ across, then $y$ up or down.
• In the x-axis, $y$ changes sign. In the y-axis, $x$ changes sign. In the origin, both change sign.
• On a horizontal line the length is $|x_2 - x_1|$. On a vertical line it is $|y_2 - y_1|$.
• The fourth corner of a parallelogram $ABCD$ is $A + C - B$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the reflection of $(2, -7)$ in the y-axis?

2. How far apart are $(4, -1)$ and $(4, 6)$?

3. Three corners of a rectangle are $(0, 0)$, $(6, 0)$ and $(6, 2)$. What is the fourth corner?

4. What is the reflection of $(-4, 5)$ in the origin?