1.3 Linear Functions

  • How did the assignment go?
  • Lab tomorrow, 11:00 am in Good Library 102. Bring your iPad with WolframAlpha

Basic Concepts

A linear function is...

  • A function with a constant rate of change: Pick any two points, and you'll always get the *same* rate of change.
  • Graph: a straight line = a constant, unchanging slope.
  • Formula:
    • output = initial value + rate of change$\times$input.
    • $f(x)=b+mx$

Table example

The table below shows the volume of liquid (in liters) in three different containers at five different times after three faucets were turned on.

  • What is $Q(20)? Interpret your answer...
  • Which of these functions is linear?
  • Graph these three functions.
  • For the linear function(s), find a formula that is consistent with the tabulated values.

Table example (cont'd)

The table below shows the volume of liquid (in liters) in three different containers at five different times after three faucets were turned on.

  • Rise/run and connection with similar triangles.
  • What are the units of the slope?

Example: formula

A stalactite grows according to the formula $$L(t)=17.75+\frac{1}{250}t, $$ where $L(t)$ is its length (in inches), and $t$ is the time (in years) since the stalactite was first measured.

  • Sketch a graph of this function.
  • Identify the vertical intercept and explain its meaning in practical terms.
  • Identify the slope and explain its meaning in practical terms. (Think aboutr units...)

Example: graph


For the blue line....

  • what's the domain of input values shown?
  • Define variables for the input and output values.
  • Find a formula for the function defined by the blue line. (Hint, find the slope first--the rate of change).
  • How do you interpret the slope and the intercept in your formula?

Image credits: Zoltan Voros