Families Of Straight Lines.
Exploring families of straight lines is a foundational skill assumed for Queensland Year 11 Mathematical Methods (QCAA). A family is a set of lines sharing a feature controlled by a parameter — a fixed gradient, or a common point they share.
You will recognise the shared feature, describe a parallel family or a fixed-point family, and find the single member that meets an extra condition — sharpening your feel for the gradient.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA), a family of straight lines is a set of lines that share a feature controlled by a parameter. A parallel family \(y=mx+c\) fixes the gradient \(m\) and varies \(c\); a fixed-point family keeps a common point and varies the gradient. This page shows how to recognise the shared feature and find the member that meets an extra condition.
A family of lines is described by an equation with a parameter. As the parameter changes, each value gives one line (a member) of the family, and all members share one feature.
In a parallel family \(y=mx+c\) the gradient \(m\) is fixed and the intercept \(c\) varies, so every member is parallel. In a fixed-point family such as \(y=mx+c_0\) (fixed \(c_0\), varying \(m\)) every member passes through the common point \((0,c_0)\); changing \(m\) rotates the line about that point, while changing \(c\) translates it up or down.
Two common families:
How to work with a family of lines
- Identify the shared feature: a fixed gradient (parallel family) or a fixed point (a family through one point).
- Write the family with its parameter, e.g. \(y=mx+c\) or \(y-y_0=m(x-x_0)\).
- Apply the extra condition (a point, an intercept, or a gradient) and solve for the parameter to name the member.
Substitute \((1,4)\) into \(y=2x+c\):
| \(4\) | \(=\) | \(2(1)+c\) |
| \(4\) | \(=\) | \(2+c\) |
Solve for the parameter \(c\):
| \(c\) | \(=\) | \(4-2\) |
| \(c\) | \(=\) | \(2\) |
Member: \(y=2x+2\).
Substitute \((3,7)\) into \(y=mx+1\):
| \(7\) | \(=\) | \(m(3)+1\) |
| \(7\) | \(=\) | \(3m+1\) |
Solve for the parameter \(m\):
| \(3m\) | \(=\) | \(6\) |
| \(m\) | \(=\) | \(2\) |
Member: \(y=2x+1\).
At the \(x\)-intercept \(y=0\) and \(x=3\); substitute:
| \(0\) | \(=\) | \(2(3)+c\) |
| \(0\) | \(=\) | \(6+c\) |
Solve for \(c\):
| \(c\) | \(=\) | \(-6\) |
Member: \(c=-6\), i.e. \(y=2x-6\).
Fixed point — the value of \(x\) that removes \(m\) is \(x=-1\):
| \(y\) | \(=\) | \(m(-1+1)+2\) |
| \(=\) | \(m(0)+2\) | |
| \(=\) | \(2\) |
So every member passes through the fixed point \((-1,\,2)\).
Substitute \((1,8)\) to find \(m\):
| \(8\) | \(=\) | \(m(1+1)+2\) |
| \(8\) | \(=\) | \(2m+2\) |
| \(2m\) | \(=\) | \(6\) |
| \(m\) | \(=\) | \(3\) |
Fixed point \((-1,\,2)\); member \(y=3x+5\).
Common pitfalls
Frequently asked questions
What is a family of straight lines?
A set of lines described by an equation with a parameter; each parameter value gives one member, and all members share one feature such as a gradient or a common point.
What does changing c do in y = mx + c?
It translates the line vertically: the gradient stays the same, so the whole family is parallel, sliding up as \(c\) increases.
What does changing m do?
It changes the steepness, rotating the line about its \(y\)-intercept \((0,c)\). Members share that point but have different gradients.
How do you find the fixed point of a family?
Write it in the form \(y-y_0=m(x-x_0)\); the fixed point is \((x_0,y_0)\), because that point makes the \(m\)-term zero for every \(m\).
How do you find the member that passes through a given point?
Substitute the point into the family equation and solve for the parameter, then write out that particular line.