Linear Function - Always a line

Definition: A linear function is a function whose graph is a line.

Regulations: A linear equation in 2 variables is a linear function.

Examples of linear equations: \bbox[border:1px #990000 dotted,1pt]{y=2x-3} \bbox[border:1px #990000 dotted,1pt]{2y+5x=2} \bbox[border:1px green dotted,1pt]{f(x)=2x-1} \bbox[border:1px green dotted,1pt]{3s+2t=2}

Regulations: There are 2 useful forms of a linear function: Slope-Intercept Form and Standard Form

Standard Form: Ax+By=C , where A, B and C are numbers More

Regulations: The standard form of a line is NOT unique.

Example: 2x-y=3 and 4x-2y=6 are the same linear function - their graph is the same line!

Slope-Intercept Form: y=ax+b , where a and b are numbers More

Regulations: The slope-intercept form of a line is unique.

Example: The line 2x-y=3 has the unique slope-intercept form: y=2x-3

InterActivity Directions for InterActivity

Look at the left in the algebra window. There are two lines a and b. Look at the drawing pad. There is only one line. That is because a and b are the same line. a: x+2y=6 and b: y=-0.5x+3 . On a piece of paper solve x+2y=6 for the variable y and see if you get y=-0.5x+3
Multiply x+2y=6 by the number 2. What is the result?
Type this result in the input field at the bottom of the interactivity and hit enter.
- You should NOT see a different line since this is the same line.
- Notice that line c: has the same numbers as line a. GeoGebra has reduced it (divided by 2).
In the input field, type -3x+y=1 and hit enter. You will get a new line.
- On a piece of paper, solve this equation for y.

Write your answer in the form y=ax+b. What is your result?

In the input field, type in this result and hit enter. You should NOT see a new line.
Think of another way to input this same line.
Input your result - the lines should coincide AND in the algebra window GeoGebra should reduce your answer so lines d and f have same numbers.