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[[1.2 Rate of Change | Prior Lesson]] | [[1.4 Finding Linear Functions | Next Lesson]]

==Objectives:==
*Determine when a function might be linear from a sample of points
*Identify the slope, x-intercept, and y-intercept of a linear function
*Explain the significance of the slope, x-intercept, and y-intercept of a linear function modeling a word problem
*Create linear functions from word problems

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;Definitions: linear function, slope, <math>x</math>-intercept, <math>y</math>-intercept

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==Lesson Guide==

===Warm-Up===

Have students do Problems 1.

===Determine when a function is linear from a sample of points===
Recall some of the problems in the 1.2 worksheet with a non-constant rate of change. Discuss with your students that linear functions always have a ''constant rate of change''.

The average rate of change of a function is usually different on different intervals. This is why linear functions are so very special. What is the definition of a linear function? Why do we call it linear? Coax the two essential points out of your students. Write them on the board.

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A '''linear function''' is a function with a constant rate of change. The graph of a linear function is a line.
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Do an example where you check if a table of points that represent a function can be linear. Explain to students that linear functions should have a constant rate of change, i.e., if we compute the average rate of change between any two pairs of points, it should always be the same. Make a point that in order to confirm a function is linear you must check every point.

{{Ex-begin}}

Draw the graph. Circle the <math>x</math>- and <math>y</math>- intercepts and ask the students if they know the names for these points. Label the points accordingly. Remind the students that the <math>y</math>-intercept occurs when the input is 0; an <math>x</math>- intercept occurs when the output is zero. We can also find the slope of the linear function by calculating the rate of change on any interval. Finally, with the initial value, <math>f(0)</math>, and the slope we can write an equation for any linear function.

''It is important to drill home how to find the <math>x</math>- and <math>y</math>-intercepts as this will be used over and over again later in the course.''

Write the following on the board and then draw the following graph.
*A linear function is a function with a constant rate of change
*The graph of a linear function is a line

[[File:Linearfunction.png|thumb]]

Output = Initial Value + Rate of Change <math>\times</math> Input

<math>f(x) = b + mx</math>

We can calculate <math>m</math> by the following formula: If <math>(x_0,y_0)</math> and <math>(x_1,y_1)</math> are two points on a line, the slope <math>m</math> is given by

<math>
m = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta y}{\Delta x} = \frac{y_1-y_0}{x_1-x_0}.
</math>

In general, to find the <math>y</math>-intercept of a function, <math>f(x)</math>, we evaluate <math>f(0)</math>. In the case <math>f(x) = mx+b</math>, we see that <math>f(0) = b</math>, so <math>(0,b)</math> is the <math>y</math>-intercept (also called the initial value).

To find the <math>x</math> intercept, we solve for <math>x</math> in <math>f(x) = 0</math>. In our case, we set <math>mx+b = 0</math>, and we see that <math>x = -\frac{b}{m}</math>.

{{Ex-end}}

Have students do Problem 3.

===Understand the terms: slope, <math>x</math>-intercept, <math>y</math>-intercept ===

Graph a linear function using a formula obtained from a table in Problem 3 that could represent a linear function. Identify the slope, the <math>x</math>-intercept, and the <math>y</math>-intercept. Discuss what these mean and write it formally on the board:

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Given a linear function <math>y=f(x),</math>
* the \underline{slope} is the the \textbf{constant} rate of change, and can be computed as <math>\frac{f(x_1)-f(x_2)}{x_1-x_2}</math> given any distinct <math>x</math> values <math>x_1</math> and <math>x_2.</math>
*The \underline{<math>x</math>-intercept} (if it exists) is the point where the function's graph crosses the <math>x</math>-axis and its <math>x</math> value is the value such that <math>f(x)=0.</math>
*The \underline{<math>y</math>-intercept} is the point where the function's graph crosses the <math>y</math>-axis and its <math>y</math> value is the value such that <math></math>y=f(0).<math></math> <math>(0,f(0))</math>
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===Create linear functions from word problems===

Do several examples where a word problem describes a rate of change which is constant and can be represented as a linear function. Emphasize that the quantity in the problem is changing is by some {\em constant} amount over some fixed unit of time, distance, etc., and this is why it can be represented by a {\em linear} function. Discuss the meanings in practical terms of the slope and <math>y</math>-intercept. Make sure you emphasize the units of each given by the context of the example.

{{Ex-begin}}
A new Toyota RAV4 costs 21,500. The car's value depreciates linearly to 11,900 in three years. Write a formula which expresses its value <math>V</math> in dollars, in terms of its age, <math>t</math>, in years.

Input: <math>t</math> = age of the car.

Output: <math>V</math> = value of the car

Initial value (<math>b</math>): \<math>21,500.

Rate of change (</math>m$): We need to calculate. (Have students remind you of the formula).

<math>
m = \frac{21500-11900}{0-3} = \frac{9600}{-3} = -3200
</math>

So <math>V = -3200t + 21500</math>.

{{Ex-end}}

Have students do Problem 4.

Have students do Problem 5.

One thing to watch for here is that it's easy to miss including the fine as a fixed cost. Also note that the slope of the linear profit function will be 12.5 dollars per shirt, not just $12.50.

==Comments==
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In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

1.3 Linear Functions - Revision history

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Determine when a function might be linear from a sample of points *Ide..."

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== Determine when a function might be linear from a sample of points Ide..."