Graphing Straight Lines
Graphing straight lines is a foundational skill assumed for Queensland Year 11 Mathematical Methods (QCAA). Because two points fix a line, a clear sketch needs only a couple of well-chosen points, such as where the line crosses each axis.
You will find the horizontal and vertical intercepts by setting the other variable to zero, sketch from the gradient and intercept, and link each line's algebraic and graphical representations with confidence.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA), sketching a straight line means plotting a couple of key points and joining them. Two reliable methods are the gradient-intercept method (plot the \(y\)-intercept, then step by the gradient) and the intercept method (plot where the line crosses each axis). This page also covers reading intercepts off an equation, and graphing horizontal and vertical lines.
The \(y\)-intercept is where a line crosses the \(y\)-axis; there \(x=0\). The \(x\)-intercept is where it crosses the \(x\)-axis; there \(y=0\). Two points fix a straight line, so a sketch needs just two well-chosen points.
From \(y=mx+c\) you can sketch quickly: plot \((0,c)\), then use the gradient \(m\) as rise over run to step to a second point. A horizontal line \(y=c\) is flat; a vertical line \(x=a\) is upright.
To find the intercepts of a line:
How to sketch a straight line
- Choose a method: from \(y=mx+c\) plot \((0,c)\) and step by the gradient; otherwise find both intercepts.
- Find two points: set \(x=0\) for the \(y\)-intercept and \(y=0\) for the \(x\)-intercept, or step off the gradient.
- Plot and rule a straight line through the points, extending it and labelling the intercepts.
Read \(m\) and \(c\) from \(y=mx+c\):
| \(m\) | \(=\) | \(2\) |
| \(c\) | \(=\) | \(-4\) |
Plot the \(y\)-intercept \((0,-4)\).
Find the \(x\)-intercept by setting \(y=0\):
| \(0\) | \(=\) | \(2x-4\) |
| \(2x\) | \(=\) | \(4\) |
| \(x\) | \(=\) | \(2\) |
Line through \((0,-4)\) and \((2,0)\), rising \(2\) for every \(1\) across.
\(x\)-intercept — set \(y=0\):
| \(3x+4(0)\) | \(=\) | \(12\) |
| \(3x\) | \(=\) | \(12\) |
| \(x\) | \(=\) | \(4\) |
\(y\)-intercept — set \(x=0\):
| \(3(0)+4y\) | \(=\) | \(12\) |
| \(4y\) | \(=\) | \(12\) |
| \(y\) | \(=\) | \(3\) |
Line through \((4,0)\) and \((0,3)\).
On the \(x\)-axis \(y=0\); substitute and solve:
| \(0\) | \(=\) | \(2x-6\) |
| \(2x\) | \(=\) | \(6\) |
| \(x\) | \(=\) | \(3\) |
\(x\)-intercept at \((3,\,0)\).
Compare the coordinates:
| \(x_1\) | \(=\) | \(2\) |
| \(x_2\) | \(=\) | \(2\) |
Both points have the same \(x\)-value, so the line is vertical. Every point on it has \(x=2\).
Equation: \(x=2\) (a vertical line, gradient undefined).
Common pitfalls
Frequently asked questions
How do you sketch a line from y = mx + c?
Plot the \(y\)-intercept \((0,c)\), then use the gradient \(m\) as rise over run to step to a second point, and rule the line.
How do you find the x-intercept and y-intercept?
Set \(y=0\) and solve for \(x\) to get the \(x\)-intercept; set \(x=0\) and solve for \(y\) to get the \(y\)-intercept.
What does the graph of y = 3 look like?
A horizontal line crossing the \(y\)-axis at \(3\); every point has \(y=3\) and the gradient is \(0\).
What does the graph of x = 2 look like?
A vertical line crossing the \(x\)-axis at \(2\); every point has \(x=2\) and the gradient is undefined.
How many points do I need to draw a line?
Two. A straight line is fixed by any two of its points, so plot two (often the intercepts) and rule through them.