Linear+Functions

//Blake Webb//


 * __Linear Functions__

Definitions**

When a linear function is written in the form //y= mx + b// and //m = 0,// then //y = b// for all //x E R.// In that case f is called a **Constant Function**, and its graph is the horizonal line through (0,b). If //b=0// and //m = 0,// we have //y = mx.// Then the function //f// is called a **direct variation**, and we say that **//y// varies directly as //x//**//,// or **//y// is directly proportional to //x//**//,// and that //m// is the **constant of variation** or the **constant of proportionality**. When the domain of a direct variation //f// is //R,// the ordered pair (0,0) is always on the graph of //f,// and hence this graph is a line passing through the origin and having slope //m.//

y1:x1 = y2:x2
 * proportion -** equality of ratios sometimes written as
 * -Means -** x1 and y2
 * -extreams -** y1 and x2


 * __Direct Variation__**

A function //f// in which the rule for pairing is given by a linear equation of the form

//y = mx + b (m, b E R)//

is called a linear function and can be written in functional notation as

//f : x//  à //mx + b//

Generally, he domain of a linear function //is// R, and its graph is a nonvertical line.

// ** Example 1 ** //- Find a rule that dofines a linear //function// G if G(8) = 7 and m = 3/4

** //Solution// //1// ** - Since m = 3/4 and (8,7) is a pair determined by G, we can substitute these values in G(x) = mx + b:

7 = 3/4 * 8 +b or b = 7 - 6 = 1

So the rule for G //is// y = 3/4x + 1 **Example 2** - Solve the linear finction f if

f(x) = 2x + 4

  **Solution 2** -    Set X = to 0

f(x)=2x+4

0=2x+4

0=-2


 * Example 3** -

F(x) = 6x - 12

0 = 6x - 12

12 = 6x


 * Solution 3** - x = 2


 * Example 4** - Word problem

When a bicycle is being pedaled in a certian gear, it travels 16m for every 3 pedal revolutions. How many revolutions would be needed to travel 600m?


 * Solution 4** - Let x = the number of pedal revolutions. Distance traveled varies directly as the number of pedal revolutions. Two ordered pairs determinded by the variation are (3,16) and (x,600). Therefore we have -

16/3 = 600x

16x = 1800

x = 112.5

It would take 112.5 revolutions to go 600m.

__**Video Aid**__

media type="youtube" key="vLxfNSnk_oA" height="344" width="425"


 * __Some helpful websites

[|North park University Math department] [|Oaklahoma State Math department Understanding Linear Functions]__**


 * __Here is a practice quiz to test your new skills__**

1. f(x) = 2x + 4

`2. f(x) = 11x - 121

3.f(x) = 4x - 42

4.f(x) = 9x + 78

Determine a rule for the linear function //f,// with the given value

5.//f// (-1) = 4; //f// (1) = -2


 * __Citations__**

Gleason, Andrew M., ed. __Algebra 2 and Trigonometry__. Boston, MA: Houghton Mufflin Company, 1992. 108-22."Solving linear functions." __Solving linear equatons__. <[|http://www.youtube.com]>.

"Linear equations." __Linear equations__. North Park University Math Department. Oct.-Nov. 2008 http://campus.northpark.edu/math/precalculus/functions/graphing/linear/index.html.

"Solving Linear Functions." __Help with solving linear equationd__. 8 Aug. 2002. Oaklahoma State University. Oct.-Nov. 2008 http://www.math.okstate.edu/~noell/ebsm/linear.html.

Quiz answers 1. x=2 2. x=11 3. x=21/2 4. x=26/3 5. y= -3x + 1

**//Last Updated: 12/4/08//**