Algebra / Linear Functions
Linear Functions
A linear function has a constant rate of change and can often be written as $f(x)=mx+b$, where $m$ is the rate of change and $b$ is the value of the function when $x=0$.
Least you need to know
- The rate of change of a linear function is constant.
- In $f(x)=mx+b$, $m$ is the slope and $b$ is the $y$-intercept.
- The slope between two points is the change in output divided by the change in input.
Key notation
- $f(x)$ — The output of the function $f$ for input $x$
- $m$ — The slope or constant rate of change of a linear function
Worked example
- Suppose $f$ is linear and $f(2)=7$ and $f(5)=19$.
- The rate of change is $\frac{19-7}{5-2}=4$.
- Since $f(x)=4x+b$, use $f(2)=7$: $7=4(2)+b$.
- Thus, $b=-1$, so $f(x)=4x-1$.
Common mistakes
- A constant difference in outputs only shows a constant rate of change when the inputs increase by equal amounts.
- The initial value is the output when the input is 0.
- Function notation such as $f(3)$ asks for an output, not multiplication.
How to recognize it
- The problem mentions a constant rate, slope, initial value, or function notation.
- A table may show equal changes in output for equal changes in input.
- The problem may ask for a value such as $f(6)$ or for an equation defining $f$.
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