Linear Functions

linear equations, intercepts and slopes

A linear equation is the equation of a line.

The standard form of a linear equation is

Ax + By = C

* A has to be positive and cannot be a fraction.

Examples of linear equations

2x + 4y =8

6y = 3 – x

47

3

x y

x + 6y = 3

The equation is in the standard form

4x - y = 21

Examples of Nonlinear Equations

4x2 + y = 5

xy + x = 5

s/r + r = 3

The exponent is 2

There is a radical in the equation

Variables are multiplied

Variables are divided

4x

The following equations are NOT in the standard form of Ax + By =C:

Determine whether the equation is a linear equation, if so write it in standard form.

32 xy

xy 25

652 yxy

354

1 yx

2x + y = 5

This is not a linear equation since its in the second degree

This is not a linear equation since variables are multiplied

x + 20y = 12

DEFINITION OF A LINEAR FUNCTION

A linear function is a function of the form

f(x) = mx + b

where m and b are real numbers and m = 0

Transform the following into the form y = mx + b

x + y = 2

2x – y = 5

8x – 2y = 12

-3x + 2y = 6

y = -x + 2

y = 2x - 5

y = 4x - 6

y = 3x + 3 2

Slope refers to the steepness of a line.

SLOPE OF A LINE

Slopes: trends

An increasing line defines a positive slope

A decreasing line defines a negative slope

A horizontal line defines a zero slope

A vertical line defines an undefined slope

Finding the slope of Linear functions

What is the slope of a line passes through points (4,6) and (3,4)?

m = 2

Determine the Slope of the following linear functions that passes through the given pair of points

1. (3, 2), (6, 6)

2. (-9, 6), (-10, 3)

3. (-4, 2), (-5, 4)

x and y intercepts

The x coordinate of the point at which the graph of an equation crosses the x –axis is the x- intercept.

The y coordinate of the point at which the graph of an

equation crosses the y-axis is called the y- intercept.

X- intercept (-x,0)

y- intercept (0, y)

3x + 2y = 9

Graph the linear equation using the x- intercept and the y intercept

To find the x- intercept, let y = 0

3x + 2y = 93x + 2(0) = 9 Replace y with 0

3x = 9 Divide each side by 3x = 3

To find the y- intercept, let x = 0

3x + 2y = 9

Replace x with 0

Divide each side by 2

3(0) + 2y = 9

2y = 9 y = 9/2

Plot the two points and connect them to draw the line.

2x + y = 4

To find the x- intercept, let y = 0Original Equation

Replace y with 0

2x + y = 4

2x + (0) = 4

2x =4 Divide each side by 3

x = 2

To find the y- intercept, let x = 02x + y = 4

2(0) + y = 4

y = 4

Original Equation

Replace x with 0

Simplify

Plot the two points and connect them to draw the line.

Find the x and y- interceptsof x = 4y – 5

● x-intercept:● Plug in y = 0

x = 4y - 5

x = 4(0) - 5

x = 0 - 5

x = -5● (-5, 0) is the

x-intercept

● y-intercept:● Plug in x = 0

x = 4y - 5

0 = 4y - 5

5 = 4y

= y

● (0, ) is the

y-intercept

5

4

5

4

Find the x and y-interceptsof g(x) = -3x – 1*

● x-intercept● ( , 0) is the

x-intercept

● y-intercept● (0, -1) is the

y-intercept

*g(x) is the same as y

1

3

Find the x and y-intercepts of x = 3

● y-intercept

● A vertical line never crosses the y-axis.

● There is no y-intercept.

● x-intercept

●There is no y.

● x = 3 is a vertical line so x always equals 3.

● (3, 0) is the x-intercept. x

y

Find the x and y-intercepts of y = -2

● x-intercept

● Plug in y = 0.

y cannot = 0 because

y = -2.● y = -2 is a horizontal

line so it never crosses

the x-axis.

●There is no x-intercept.

● y-intercept

● y = -2 is a horizontal line

so y always equals -2.

● (0,-2) is the y-intercept.

x

y

EQUATION OF A LINEAR FUNCTION

Slope- Intercept form

y = mx + b

y = mx + b

Give the equation of the linear function y in slope intercept form given its slope and y-intercept

1. m = -3, b = 2

2. m= 2, b = - 4

3. M = 1/3, b = 3

EQUATION OF A LINEAR FUNCTION

Point-Slope form

y –y1= m(x – x1)

y –y1= m(x – x1)

Give the equation of the linear function y with the given slope and passing through given points.

1. m = 2, through (1, 2)

2. m= -3, through (5, 0)

3. m = -1/3, through (-1, 3)

EQUATION OF A LINEAR FUNCTION

Give the equation of the linear function y with the given slope and passing through given points.

1. through (1, 2) and (3, -2)

2. through (5, 0) and (-1, 3)

Intercept Form

_x_ + _y_

a b

EQUATION OF A LINEAR FUNCTION

= 1

Questions??