Plotting Straight-Line Graphs
Straight-line graphs show a steady relationship between two variables and are used everywhere, from calculating your phone bill to plotting scientific data. This guide will walk you through the basics of plotting these lines and understanding what their equations tell you.
How to Plot a Straight-Line Graph
The simplest way to plot a graph is by creating a table of values. You choose some $x$ values, plug them into the equation to find their matching $y$ values, and then plot the $(x,y)$ coordinate pairs.
Step-by-Step Guide to Plotting
- Start with the equation (e.g., $y = 2x + 1$).
- Create a table of values for $x$ and $y$.
- Choose at least three $x$ values (e.g., −1, 0, 1). A third point checks for errors!
- Calculate the corresponding $y$ values by substituting each $x$ into the equation.
- Plot the $(x,y)$ coordinate pairs on your grid.
- Use a ruler to draw a straight line through all the points.
Understanding the Equation: $y = mx + c$
The general equation for any straight line tells you two key things at a glance.
The Two Key Parts
‘m’ is the Gradient: This tells you how steep the line is.
- Positive m: Line slopes upwards (uphill).
- Negative m: Line slopes downwards (downhill).
- The bigger the number, the steeper the line.
‘c’ is the Y-intercept: This is the point where the line crosses the vertical y-axis. Its coordinate is always $(0, c)$.
Worked Example: Plotting $y = 2x – 3$
Step 1 & 2: Create and complete a table of values.
| $x$ | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| $y$ | −7 | −5 | −3 | −1 | 1 |
Calculations: When $x = -2$: $y = 2(-2) – 3 = -7$. When $x = 0$: $y = 2(0) – 3 = -3$. When $x = 2$: $y = 2(2) – 3 = 1$.
Step 3: Plot the points and draw the line.
From the equation $y = 2x – 3$, we can also read off that the gradient is 2 and the y-intercept is −3, which matches our table perfectly!
Tutor Insights
🤔 Common Misunderstandings
- Swapping X and Y: Plotting coordinates as $(y, x)$ instead of $(x, y)$.
- Negative Signs: Making calculation errors with negative numbers when filling in the table of values.
- Gradient Confusion: Thinking a negative gradient means the line is flat, or that a larger number means a less steep line.
📝 Common Exam Mistakes
- Inaccurate Plotting: Not using a ruler or plotting points slightly off the grid lines.
- Not Labelling: Forgetting to label the axes or the line with its equation.
- Failing to Rearrange: Trying to identify ‘m’ and ‘c’ from an equation like $2y = 4x + 6$ without first rearranging it to $y = 2x + 3$.
Practice Questions
- Plot the graph of $y = x + 2$ for $x$ values from −3 to 3.
- For the equation $y = 3x – 5$, identify the gradient and the y-intercept.
- Rearrange $x + y = 7$ into the form $y = mx + c$, then state the gradient and y-intercept.
- A line has a gradient of 3 and crosses the y-axis at $(0, -2)$. Write down the equation of this line.
Show Answers
- Your table should include points such as $(−3, −1)$, $(0, 2)$, and $(3, 5)$. Draw a straight line through all plotted points.
- Gradient ($m$) = 3. Y-intercept ($c$) = −5.
- Rearranged: $y = -x + 7$. Gradient ($m$) = −1. Y-intercept ($c$) = 7.
- $y = 3x – 2$
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