FUNCTIONS 04
Linear Functions
A linear function changes at a constant rate, so equal changes in the input produce equal changes in the output.
The main idea
A linear function has the form
\[\boxed{f(x)=mx+c}.\]The number \(m\) controls the slope of the line, while \(c\) tells us where it crosses the \(y\)-axis.
1. Why the graph is a straight line
In \(f(x)=mx+c\), every increase of one unit in \(x\) changes the output by the same amount \(m\).
Because the rate never changes, the graph does not bend.
2. The gradient \(m\)
The gradient measures vertical change per unit horizontal change:
\[\boxed{m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}}.\]For \(f(x)=2x+1\), moving \(3\) units to the right produces a rise of \(6\), so \(m=6/3=2\).
3. The intercept \(c\)
At \(x=0\),
\[f(0)=m(0)+c=c.\]So \(c\) is the \(y\)-coordinate where the line crosses the vertical axis.
For \(f(x)=2x+1\), the intercept is \((0,1)\).
4. Positive, negative and zero gradient
If \(m>0\), the function increases. If \(m<0\), it decreases. If \(m=0\), the output is constant.
5. Finding a line from two points
Suppose a line passes through \((1,3)\) and \((4,9)\).
\[m=\frac{9-3}{4-1}=2.\]Using \(y=mx+c\) and the point \((1,3)\),
\[3=2(1)+c\Rightarrow c=1.\]Hence \(f(x)=2x+1\).
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find \(m\)
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substitute one point
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find \(c\)
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\(f(x)=mx+c\)
6. Parallel lines
Lines with the same gradient have the same direction and never meet.
For example, \(y=2x+1\) and \(y=2x-3\) both have gradient \(2\), but different intercepts.
7. Linear models
Linear functions are useful when one quantity changes by a fixed amount for every unit increase in another.
For example,
\[C(n)=5n+20\]can represent a fixed charge of \(20\) plus \(5\) for each unit.
\(+5\) for each unit
\(C(0)=20\)
\(C(1)=25\)
\(C(2)=30\)
8. Average rate is the same everywhere
For a linear function, any two distinct points give the same average rate of change.
\[\frac{f(x_2)-f(x_1)}{x_2-x_1}=m.\]This is special: for curved functions, the corresponding slope generally depends on which two points are chosen.
9. From straight lines to curves
A straight line has one slope everywhere. A curve generally has a changing slope.
Calculus asks for the slope of a curve at a particular point by using a tangent line.
A linear function has the form \(f(x)=mx+c\). Its gradient \(m\) is a constant rate of change, and \(c\) is its vertical intercept. Equal changes in input always produce proportional changes in output.
Why this matters for calculus
The gradient of a straight line is the simplest example of a rate of change. Differentiation generalises this idea: instead of one constant slope for an entire line, it finds the local slope of a smooth curve at each point.