Linear Functions
But first...
CoCalc tips
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- ...or, guess and try something.
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- When lab is due (following Thursday evening), your entire 'assignment-01' folder gets copied to me: No need to 'hand in' anything.
- Assume $y=100-2x$. If $x$ goes up by 3, the corresponding $y$-value changes by
(a) 300 (b) -300 (c) 6 (d) -6 (e) 94 (f) -94
show / hideThe slope, $m=-2$ is the rate of change of the output when you change the input by 1 unit.
We're changing the input by 3 units. So the change of the function is $3m=3*(-2)=$-6. Answer (d). - Equation of this line?
Use the $y$-intercept, (0,6), and the $x$-intercept, (2,0), as the two points to find the slope: $$m=\frac{\Delta y}{\Delta x}=\frac{0-6}{2-0}=-frac{-6}{2}=-3.$$ The $y$-intercept, 6, for the common equation of the line is "$b$". So the equation is $$y=mx+b=\color{blue}{-3x+6}.$$ - Which of the following lines have the same slope?
- $y=3x+2$
- $3y=9x+4$
- $3y=2x+6$
- $2y=6x+4$
Using algebra, equations a), b), and d) can all be rearranged to $y=3x+...$. (The slope of equation c) is 2/3).
Shock
verbal When a person goes into shock, their cardiac output, in liters of blood per minute, decreases. One person's output is 12 liters per minute when she first goes into shock, and decreases by 2 liters per minute every hour she is in shock.
- table Construct a table for her cardiac output for the first 5 hours
- graph Graph these points
- analytical What is the equation for the line through these points?
Vertical intercept is 12, slope is -2... $$\color{blue}{o(t)=12-2t}$$
- Linear data?
In table (b) the output changes by -3 each time the input changes by 1.
None of the others are linear. For example, in table (d), the output is changing by 2 in each line of the table, but the input is changing first by 1, then by 2...
| time (hrs) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| output (l/min) | 12 | 10 | 8 | 6 | 4 |
[1.3] Average rate of change
From a table
This table gives sales of the medicinal herb saw palmetto, in millions of dollars, for several years.
| Year | 1997 | 1998 | 1999 | 2000 | 2001 |
|---|---|---|---|---|---|
| Sales (million $) | 85 | 107 | 116 | 122 | 123 |
The average rate of change between 1997 and 2001 was:
$$\frac{\Delta y}{\Delta x}=\frac{123-85}{2001-1997}
=\frac{38}{4}=9.5 \text{ million \$ per year}$$
(b) increased by an average of 9.5 million \$ each ("per") year
Between $x=1$ and $x=3$. positive (increasing)
Between $x=4$ and $x=8$.negative (decreasing)
Between $x=4$ and $x=6.$\approx 0$ (flat): $y(4)$ and $y(6)$ are approximately the same.
Which of the following graphs might represent the position of an object which is slowing down? Speed is the instantaneous slope on a position vs time graph. Both (a) and (c) start shallow (slow) and become steep (fast).
But graph (b) starts steep (fast) and becomes shallower (slower).
(true or false) The graph of any function is always either concave up or concave down.
Say for each interval whether the average rate of change is positive, negative, or zero.
Motion
Concavity