0580 · 3.2Drawing linear graphs

Plotting points for a linear graph

0:00An equation can describe a whole line, not just one answer. Every point on that line gives a pair of values that makes the equation true. Drawing the graph turns an algebraic rule into something you can see. By the end, you will understand how to calculate, plot and check these points. You need two starting skills: substituting into an expression and locating coordinates. The horizontal axis shows x, and the vertical axis shows y.

0:24For the point with x equal to two and y equal to one, move right two units, then up one. Negative coordinates take you left or down from the origin. A graph of an equation shows all the points whose coordinates satisfy it. That means substituting their values makes the two sides equal. For example, y = −2x + 5 connects each chosen x value to a corresponding y value.

0:47We calculate several pairs, then plot their positions. These points lie on a straight line because equal increases in x produce equal changes in y. In this example, increasing x by one decreases y by two. That pattern continues between the points we calculate, not just at whole numbers. Joining them therefore represents all the intervening solutions too. Our first worked question is to draw y = −2x + 5.

1:13Choose x values of negative one, zero and two. For negative one, substitute to get negative two multiplied by negative one plus five. The multiplication gives positive two. Adding five gives y equal to seven. When x is zero, y equals negative two multiplied by zero plus five. The multiplication gives zero, so y = 5. For x equal to two, calculate negative two multiplied by two plus five. This becomes −4 + 5, giving y equal to one.

1:44Plot the point with x equal to negative one and y equal to seven. Next, mark the points zero comma five and two comma one. Place a ruler through all three and draw one continuous straight line. Unless the question restricts the range, extend it across the available grid rather than stopping at your points. Two distinct points determine a straight line, but a third provides a useful check. If it does not lie on the same line, inspect your arithmetic and plotting.

2:09Do not bend the line to include an incorrect point. A straight line cannot change direction halfway across the grid.

Worked examples with different equations

2:15The second question is to draw y = 7 − 4x. For x equal to zero, substitute to obtain seven minus four multiplied by zero. Multiplication gives zero, leaving 7 − 0. So the first y value is seven. With x equal to one, calculate seven minus four multiplied by one. This becomes 7 − 4, giving three. For x equal to two, the expression is seven minus four multiplied by two.

2:41Work out the multiplication first to obtain 7 − 8. The result is negative one. Plot zero comma seven, one comma three, and two comma negative one. Draw a ruled straight line through them. Although this equation puts seven first, the plotting method has not changed. Calculate matching coordinates rather than relying on how the equation looks. Our third exam-style question is to draw 3x + 2y = 5. We can substitute directly, even though y is not alone.

3:08Choosing x equal to negative one gives −3 + 2y = 5. Add three to both sides to obtain 2y = 8. Divide by two to get y equal to four. For x equal to one, substitution gives 3 + 2y = 5. Subtract three from both sides, leaving 2y = 2. Dividing by two gives y equal to one. Each operation keeps the equation balanced while finding the matching vertical coordinate.

3:34Choose x equal to three for the final point. Substitution gives 9 + 2y = 5. Subtract nine to obtain 2y = −4. Dividing both sides by two gives y equal to negative two. Plot negative one comma four, one comma one, and three comma negative two. Draw a straight line through all three points. These chosen x values gave whole-number answers, which are convenient to plot. Fractional coordinates are also valid.

4:00The graph includes more than just whole-number points.

Graph accuracy and common mistakes

4:02Common mistakes include reversing coordinates, losing negative signs and misreading the scale. Reversing coordinates changes the point because horizontal and vertical positions have different jobs. Keep equal numerical intervals equally spaced on each axis. The two axes may use different scales, but each must be consistent. In an examination, draw means produce an accurate graph, not merely a rough sketch. Show your calculated values, plot carefully and use a ruler.

4:30These calculations must work without a calculator on Paper two. Paper four permits one. Keep exact values here, since unnecessary rounding can move points away from the correct line. The central idea is that a line collects every coordinate pair satisfying its equation. Choose convenient x values, calculate the corresponding y values, then plot them accurately. A third point checks your work before you draw the straight line. The graph is the equation made visible.