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Coordinates and Graphs

Geometry · Coordinates and Graphs

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.1 Coordinates and Graphs (6 Lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

  • Plot points on the Cartesian plane using ordered pairs.
  • Generate a table of values and draw the graph of a linear equation.
  • Solve simultaneous linear equations graphically.
  • Interpret graphs that represent real-life relationships.

Coordinates and Graphs

A point on a map needs two numbers to fix it. So does a point on a graph. Once you can plot points, you can draw any straight line and read information straight off it.

a) The Cartesian plane

Two number lines crossing at right angles make a Cartesian plane. The across line is the x-axis and the up line is the y-axis. They meet at the origin.

The four quadrants The four quadrants x y −5 −4 −3 −2 −1 1 2 3 4 5 −4 −2 2 4 0 I II III IV (3, 2)

A point is written as a pair, x first and y second. The point (3, 2) means three across and two up.

The order matters. (3, 2) and (2, 3) are different points.

The axes cut the plane into four quadrants, numbered anticlockwise from the top right. In quadrant I both numbers are positive. In quadrant III both are negative.

b) Drawing a straight line graph

To draw a line from its equation, make a table of values first.

Building a table of values Building a table of values y = 2x + 1 the equation 2(0) + 1 = 1 so y is 1 when x is 0 2(1) + 1 = 3 so y is 3 when x is 1 2(2) + 1 = 5 so y is 5 when x is 2
Plotting y = 2x + 1 Plotting y = 2x + 1 x y −2 −1 1 2 3 4 5 −2 2 4 6 0 (0, 1) (2, 5)

Choose three points, not two. Two points always make a line, even if one of them is wrong. The third point catches the mistake, because it will not sit on the same line.

Use a ruler and extend the line past your plotted points.

c) Parallel lines

Parallel lines never meet. On a graph they climb at exactly the same rate.

Parallel lines Parallel lines x y −1 1 2 3 4 5 6 −1 1 2 3 4 5 6 0

That gives a simple test. Parallel lines have equal gradients.

So y = 3x + 1 and y = 3x − 4 are parallel. Both have gradient 3. The different c values just shift one above the other.

d) Perpendicular lines

Perpendicular lines cross at a right angle.

Perpendicular lines Perpendicular lines x y −1 1 2 3 4 5 6 −1 1 2 3 4 5 6 0

The rule here is less obvious. Turn the gradient upside down, then change its sign.

Perpendicular gradients Perpendicular gradients first gradient = 2 the line you are given turn it over = 1 / 2 the reciprocal change the sign = −1 / 2 the perpendicular gradient check → 2 × −½ = −1 the product is always −1

There is a quick check. Multiply the two gradients together. If the answer is −1, the lines are perpendicular.

A gradient of 3 pairs with −⅓. A gradient of −2 pairs with ½.

e) Where this is used

A surveyor plots plot corners on a grid. A shopkeeper graphing daily sales spots a trend from the gradient. Builders use the perpendicular rule to set out square corners.

Words to know

  • Cartesian plane -- the flat surface formed by two perpendicular number lines used to locate points.
  • Ordered pair -- two numbers written as x then y that fix a point on the plane.
  • Origin -- the point where the two axes cross, with coordinates (0, 0).
  • Quadrant -- one of the four regions the axes divide the plane into.
  • Simultaneous equations -- two or more equations satisfied by the same values of the unknowns.

:::checkpoint Check yourself

  1. Which quadrant contains the point (−4, 3)?
  2. Complete the table for y = 3x − 2 when x is 0, 1 and 2.
  3. What is the gradient of a line parallel to y = 5x + 7?
  4. What is the gradient of a line perpendicular to y = 4x − 1? :::

Bridge to practice

The exercises begin with identifying quadrants and reading coordinates, move through generating tables of values, and finish with graphical solution of simultaneous equations and interpretation of real-life graphs. When a question asks for a graphical solution, always substitute your answer back into both equations; that check is quick and catches misreadings of the scale.

Check yourselfPractise Coordinates and Graphs — 10 questions →Next in MathematicsArea →