[1.4] Formulas for Linear Functions
Note on Homework: When you're asked for an exact answer, that often means that you should enter a fraction: e.g. "1/3", and not "0.33333".
Questions on WileyPlus / Lab 01 (Due on Monday)?
Basic concepts
What is a linear function?
- A function with a constant rate of change.
- Graph:
- A straight line, or points that all lie on a straight line.
- The line has a slope = rate of change.
- The line has a vertical intercept at the "initial value".
- Formula:
- Output = initial value + rate of change $\times$ input.
- $f(x)=b+mx$
Finding the formula
The equation for a line has
- an independent variable, $x$,
- a dependent variable, $f(x)$,
- and 2 parameters (or constants), $b$ and $m$.
You need 2 pieces of information to find the 2 parameters.
Finding the equation of a line from two points
A line runs through two points, (1,8) and (3,16). Find the equation of the line.
Step 1: find the rate of change, $m$ using the two points: $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{16-8}{3-1}=\frac{8}{2}=4.$$
Step 2: Substitute one point into the equation of your line and then solve for $b$.
Let's use the second point (3,16), but either point will work. This point means that $f(3)=16$. $$\begineq f(x)&=&b+m*x\\ 16 &=& b+4*(3)\\ 16&=&b+12. \endeq$$
Now solve this equation for $b$ to find... $$b=16-12=4$$
Substitute $m$ and $b$ back into the equation of a line $$\color{red}{f(x)=4+4x}.$$
If your two pieces of information are, for example, $m$ and one other point, then you can go straight to step 2.