Drawing linear graphs

1. Overview

Drawing linear graphs is a core skill in IGCSE Mathematics. It's about visually representing equations of the form $y = mx + c$ as straight lines on a coordinate plane. This skill is crucial for solving simultaneous equations graphically, understanding rates of change, and modelling real-world relationships. You'll need to be accurate with plotting points and using a ruler to draw straight lines.

Key Definitions

• Linear Equation: An equation where the highest power of the variable is 1 (e.g., $y = 2x + 3$). It always results in a straight line when graphed.
• Gradient ($m$): A measure of the steepness of a line, calculated as the "rise over run" or the change in $y$ divided by the change in $x$.
• $y$-intercept ($c$): The point where the graph crosses the $y$-axis (where $x = 0$).
• Table of Values: A systematic list of $x$-coordinates and their corresponding $y$-coordinates used to plot a graph.
• Origin: The point $(0, 0)$ where the $x$ and $y$ axes intersect.

Core Content

To draw a linear graph, you usually follow one of two main methods: the Table of Values method or the Gradient-Intercept method.

Method 1: The Table of Values (Recommended for Accuracy)

This is the most reliable method for IGCSE exams. Even though a straight line is uniquely defined by two points, calculating three points acts as a check for accuracy. If the three points do not lie on a straight line, you have made a calculation error.

Step-by-step:

1. Rearrange the equation into the form $y = ...$ if it isn't already.
2. Choose three simple values for $x$ (usually $-2, 0, 2$ or $0, 1, 2$).
3. Substitute these $x$ values into the equation to find the $y$ values.
4. Plot the coordinates on the grid.
5. Join the points with a long, straight line using a ruler.

Worked example 1 — Drawing $y = 2x - 1$

Question: Draw the graph of $y = 2x - 1$ for $-2 \leq x \leq 2$.

• Step 1: Calculate $y$ values

  • When $x = -2$: $y = 2(-2) - 1$ $y = -4 - 1$ $y = -5$. Coordinate: $(-2, -5)$
  • When $x = 0$: $y = 2(0) - 1$ $y = 0 - 1$ $y = -1$. Coordinate: $(0, -1)$
  • When $x = 2$: $y = 2(2) - 1$ $y = 4 - 1$ $y = 3$. Coordinate: $(2, 3)$

• Step 2: Table of Values

  $x$	-2	0	2
  $y$	-5	-1	3

• Step 3: Plot and Draw

  📊A Cartesian coordinate grid showing three points at (-2, -5), (0, -1), and (2, 3) connected by a solid straight line that extends slightly past the outer points.

Worked example 2 — Drawing $y = -x + 3$

Question: Draw the graph of $y = -x + 3$ for $-3 \leq x \leq 3$.

• Step 1: Calculate $y$ values

  • When $x = -3$: $y = -(-3) + 3$ $y = 3 + 3$ $y = 6$. Coordinate: $(-3, 6)$
  • When $x = 0$: $y = -(0) + 3$ $y = 0 + 3$ $y = 3$. Coordinate: $(0, 3)$
  • When $x = 3$: $y = -(3) + 3$ $y = -3 + 3$ $y = 0$. Coordinate: $(3, 0)$

• Step 2: Table of Values

  $x$	-3	0	3
  $y$	6	3	0

• Step 3: Plot and Draw

  📊A Cartesian coordinate grid showing three points at (-3, 6), (0, 3), and (3, 0) connected by a solid straight line that extends slightly past the outer points.

Method 2: Gradient-Intercept ($y = mx + c$)

1. Identify the $y$-intercept ($c$) and plot this point on the $y$-axis.
2. Use the gradient ($m$) to find the next point. If $m = 3$, move 1 unit right and 3 units up. If $m = -\frac{1}{2}$, move 2 units right and 1 unit down.
3. Connect the points.

Worked example 3 — Using gradient-intercept to draw $y = -\frac{1}{2}x + 3$

Question: Draw the graph of $y = -\frac{1}{2}x + 3$.

• Intercept ($c$): $+3$. Plot point at $(0, 3)$.
• Gradient ($m$): $-\frac{1}{2}$. From $(0, 3)$, move right 2 units and down 1 unit to reach $(2, 2)$.
• Draw: Join $(0, 3)$ and $(2, 2)$ with a ruler.

Worked example 4 — Using gradient-intercept to draw $y = 3x - 2$

Question: Draw the graph of $y = 3x - 2$.

• Intercept ($c$): $-2$. Plot point at $(0, -2)$.
• Gradient ($m$): $3 = \frac{3}{1}$. From $(0, -2)$, move right 1 unit and up 3 units to reach $(1, 1)$.
• Draw: Join $(0, -2)$ and $(1, 1)$ with a ruler.

Horizontal and Vertical Lines

• Vertical Lines: Have the equation $x = k$ (where $k$ is a constant). Every point on the line has the same $x$-coordinate.
• Horizontal Lines: Have the equation $y = k$ (where $k$ is a constant). Every point on the line has the same $y$-coordinate.

Extended Content (Extended Only)

While drawing basic linear graphs is a Core skill, understanding how to manipulate equations to get them into a suitable form for graphing can be considered an extension of this skill. This involves rearranging equations and dealing with more complex coefficients. For example, you might be given an equation like $2y + 4x = 6$ and need to rearrange it into the form $y = mx + c$ before you can easily graph it.

Worked example 5 — Rearranging and graphing $2y + 4x = 6$

Question: Rearrange the equation $2y + 4x = 6$ into the form $y = mx + c$, and then draw the graph.

• Step 1: Isolate the term with $y$ $2y + 4x = 6$ $2y = -4x + 6$ (Subtract $4x$ from both sides)

• Step 2: Solve for $y$ $2y = -4x + 6$ $y = \frac{-4x + 6}{2}$ (Divide both sides by 2) $y = -2x + 3$

• Step 3: Identify the gradient and y-intercept Now the equation is in the form $y = mx + c$, where $m = -2$ (the gradient) and $c = 3$ (the y-intercept).

• Step 4: Draw the graph using the gradient-intercept method

  • Plot the y-intercept at $(0, 3)$.
  • Use the gradient of $-2 = \frac{-2}{1}$ to find another point. From $(0, 3)$, move 1 unit right and 2 units down to reach $(1, 1)$.
  • Draw a straight line through these two points.

Key Equations

• General Equation of a Straight Line: $\bf{y = mx + c}$
  • $m$ = Gradient (steepness)
  • $c$ = $y$-intercept
• Gradient Formula: $\bf{m = \frac{y_2 - y_1}{x_2 - x_1}}$ (must be memorized)

Common Mistakes to Avoid

• ❌ Wrong: Drawing a freehand line. ✓ Right: Always use a ruler for linear graphs. A freehand line, even if mostly straight, will lose marks.
• ❌ Wrong: Calculating only two points and making an error in one. ✓ Right: Calculate and plot at least three points. If they don't align perfectly, you know you've made a mistake and can check your working.
• ❌ Wrong: Ignoring the sign of the gradient. ✓ Right: A positive gradient ($y = 2x$) must go "uphill" from left to right, and a negative gradient ($y = -2x$) must go "downhill" from left to right. Double-check this visually after drawing your line.
• ❌ Wrong: Drawing a short line segment instead of extending it across the grid. ✓ Right: Draw the line so it fills the provided grid or covers the requested range of $x$ values. Don't just stop at the points you plotted.
• ❌ Wrong: Forgetting to rearrange the equation before plotting. ✓ Right: If you're given an equation like $2y + x = 4$, rearrange it to the form $y = mx + c$ first before attempting to plot any points.

Exam Tips

• Command Words: Look for "Draw" (requires high accuracy) versus "Sketch" (requires general shape and key points labeled).
• Calculator Tip: Most modern calculators have a TABLE mode. Input your equation ($f(x) = ...$), set your "Start," "End," and "Step" (usually 1), and it will generate your coordinates for you instantly. Use this to check your manual calculations.
• Check your scale: IGCSE papers often use different scales for $x$ and $y$ axes (e.g., 2cm = 1 unit on $x$, but 1cm = 1 unit on $y$). Check carefully before plotting.
• Label your lines: If you are asked to draw more than one line on the same grid, label each line with its equation to avoid losing marks for clarity.
• Sharpen your pencil: Points plotted with a thick, blunt pencil may lead to inaccuracies that fall outside the "tolerance" allowed by markers (usually $\pm 1$mm).
• Double-check the y-intercept: After drawing your line, visually confirm that it crosses the y-axis at the value of 'c' in your equation ($y = mx + c$). This is a quick way to spot errors.