23 terms

Intercept

Is a point where a line crosses the x or y axis. Has the form (x,0) or (0,y).

How to calculate the x intercept for a linear equation?

Substitute 0 in for y in the equation and then solve for x.

Ex: Draw Graph

2x-3y=6

2x-3(0)=6

2x=6

2x/2 6/2

x=3 (3,0)

Ex: Draw Graph

2x-3y=6

2x-3(0)=6

2x=6

2x/2 6/2

x=3 (3,0)

How to calculate the y intercept for a linear equation?

Substitute 0 in for x in the equation and then solve for y.

Ex: Draw Graph

2x-3y=6

2(0)-3y=6

-3y=6

-3y/-3 6/-3

y=-2 (0,-2)

Ex: Draw Graph

2x-3y=6

2(0)-3y=6

-3y=6

-3y/-3 6/-3

y=-2 (0,-2)

Graphing a line using x and y intercepts

You can graph a line using x and y intercepts together.

Ex:

y=3x+6 Draw Graph

0=3x+6

-6 -6

-6=3x

x=-2

y=3(0)=6

y=6

x int=-2 (-2,0)

y int=6 (0,6)

Ex:

y=3x+6 Draw Graph

0=3x+6

-6 -6

-6=3x

x=-2

y=3(0)=6

y=6

x int=-2 (-2,0)

y int=6 (0,6)

What is slope?

Slope is a number which describes the ratio of the change in y to the change in x in the graph of a line.

Rise=y change y

Run =x change x

Rise=y change y

Run =x change x

Positive slope

The line rises from left to right. Larger slopes are more vertical and are more positive. Smaller slopes are more horizontal and are less positive.

Negative slope

The slope falls or goes down from left to right. They can not be both negative at the same time. They can be -,+ or +,-. The larger slopes are more vertical and are more negative. Smaller slopes are more horizontal and less negative.

Calculating slopes

(x1,y1) (x2,y2) then put into slope formula

Slope formula

m=y2-y1

x2-x1

Ex: Draw Graph

(-2,4) (6,-1)

m=-1-4

6--2

m=-5

8

x2-x1

Ex: Draw Graph

(-2,4) (6,-1)

m=-1-4

6--2

m=-5

8

Slope of horizontal line

A horizontal line has a slope of 0. The y will always be the same.

Ex: Draw Graph

3-3=0

2+4

6,0=0

Ex: Draw Graph

3-3=0

2+4

6,0=0

Slope of vertical line

A vertical line has a slope of undefined, none, or no slope. The x will always be the same.

Ex: Draw Graph

-1-3

-4--4= undefined

0,-4=none

Ex: Draw Graph

-1-3

-4--4= undefined

0,-4=none

Slope of parallel line

Parallel lines have equal slopes.

Ex:

m=2,1

||m=2,1

Ex:

m=2,1

||m=2,1

Slope of perpendicular line

Perpendicular lines have opposite signed reciprocal.

Ex:

m=2,1

|-m=1,-2

Ex:

m=2,1

|-m=1,-2

Slope form of a line and graphing with slope form

You can use the slope formula. To find the missing values in ordered pairs write the "slope form of an equation".

Ex: Find the slope of the missing coordinate.

m=3,2 (x,4) (2,-3)

2 -3-4

=

3 2-3

-21=4-2x

-4 -4

-25 -2x

=

-2 -2

x=25

2

Ex: Find the slope of the missing coordinate.

m=3,2 (x,4) (2,-3)

2 -3-4

=

3 2-3

-21=4-2x

-4 -4

-25 -2x

=

-2 -2

x=25

2

Slope form

y-y1

=m

x-x1

y and x stay as variables. y1,x1, and m are numbers

Ex: Find X

m=3,1 (x,-2) (4,5)

1 5+2

=

3 4-x

21=4-x

-4 -4

17=-x

-1 -1

x=-17

=m

x-x1

y and x stay as variables. y1,x1, and m are numbers

Ex: Find X

m=3,1 (x,-2) (4,5)

1 5+2

=

3 4-x

21=4-x

-4 -4

17=-x

-1 -1

x=-17

Graphing lines from slope form

If you have slope form then (x,y) is a point on the line and m is the slope. You can draw the graph based on these numbers.

Ex: Graph and Draw a Graph

y-2 3

=

x+1 2

(-1,2)

Ex: Graph and Draw a Graph

y-2 3

=

x+1 2

(-1,2)

Deriving standard form form slope form

If you have slope form, you may derive standard form by:

a. first cross multiply

b. then using opposite operations

to get the x and y terms on the left side of the equation and the constant term on the right side.

a. first cross multiply

b. then using opposite operations

to get the x and y terms on the left side of the equation and the constant term on the right side.

Standard form

Standard form (Ax+By=C) requires that A,B,C are integers. A is positive and A,B, and C do not have common factors other than 1. 2x+4y+8--->x+2y=4

Ex: Write the following equation in standard form.

y+2 1

=

x-3 3

x-3=3y+6

-3y +3 -3y +3

x-3y=9

*Do the x cross product first

*If you have a negative slope put the negative sign on the bottom

Ex: Write the following equation in standard form.

y+2 1

=

x-3 3

x-3=3y+6

-3y +3 -3y +3

x-3y=9

*Do the x cross product first

*If you have a negative slope put the negative sign on the bottom

Point slope form

y-y1=m(x-x1)

You need a point (x,y) and a slope to write point slope form.

Ex:

(-4,6) m=-2/3

y-6=-2/3(x+4)

You need a point (x,y) and a slope to write point slope form.

Ex:

(-4,6) m=-2/3

y-6=-2/3(x+4)

Slope intercept form

y=mx+b

m is the slope

(0,b) b is the y intercept

To put any equation in slope intercept form ISOLATE Y

Ex: Find the slope and the y intercept and draw the graph

y=6x+3

m=6/1 (0,3)

Write the equation in slope intercept form for the line that passes through (-2,1) and has a slope of 1/3. *solve for b

1=1/3(-2)+b

1=-2/3+b

+2/3 +2/3

b=5/3

y=1/3x+5/3

m is the slope

(0,b) b is the y intercept

To put any equation in slope intercept form ISOLATE Y

Ex: Find the slope and the y intercept and draw the graph

y=6x+3

m=6/1 (0,3)

Write the equation in slope intercept form for the line that passes through (-2,1) and has a slope of 1/3. *solve for b

1=1/3(-2)+b

1=-2/3+b

+2/3 +2/3

b=5/3

y=1/3x+5/3

Forms

Ax+By=C Standard form

y-y1 m

= Slope Form

x-x1 1

y-y1=m(x-x1) Point Slope Form

y=mx+b Slope Intercept Form

y-y1 m

= Slope Form

x-x1 1

y-y1=m(x-x1) Point Slope Form

y=mx+b Slope Intercept Form

Linear equation word problems

Real world situations can be represented by linear equations.

Ex: Outside temperature in NC

Independent X---> explanatory dependent y--> response

Ex:

Month 1: Average Temp-36F

Electricity cost-$68

Month 8: Average Temp-82F

Electricity cost-$140

Write ordered pairs (36,68) and (82,140) use pairs to write slope intercept form

140-68 36 cost

=

82-36 23 temperature

68=36/23(36)+b

68=1296/23+b

268/32=b

y=36/23+268/23

a. Write the slope intercept form of the equation of the line through the ordered pairs (36 temp, 68 cost) and (82 temp, 140 cost).

b.What is the slope of the equation?

c. Interpret the slope in the context of the problem.

d. What is the y intercept of the equation?

e. What does the y intercept represent?

f. Is the y intercept representation reasonable? Why?

g. When the cost is x what will the temperature be?

Ex: Outside temperature in NC

Independent X---> explanatory dependent y--> response

Ex:

Month 1: Average Temp-36F

Electricity cost-$68

Month 8: Average Temp-82F

Electricity cost-$140

Write ordered pairs (36,68) and (82,140) use pairs to write slope intercept form

140-68 36 cost

=

82-36 23 temperature

68=36/23(36)+b

68=1296/23+b

268/32=b

y=36/23+268/23

a. Write the slope intercept form of the equation of the line through the ordered pairs (36 temp, 68 cost) and (82 temp, 140 cost).

b.What is the slope of the equation?

c. Interpret the slope in the context of the problem.

d. What is the y intercept of the equation?

e. What does the y intercept represent?

f. Is the y intercept representation reasonable? Why?

g. When the cost is x what will the temperature be?

Identifying Linear Equations

The standard form of a linear equation is Ax+By=C

A,B, and C are real numbers, A and B cannot both be 0 at the same time, and A must be positive. The exponents of x and y are ones. The graph of these equations always plots in a straight path.

Ex:

Linear=4x+3y=5

Nonlinear=4x2+3y=5

y=n always plots as a horizontal line

y=3

x=n always plots as a vertical line

x=4

Graphing a linear equation by plotting ordered pair solutions. Use a function table to derive ordered pairs

Ex:

y=-2x+7

x | -2x+7 | y

-1 | -2(-1)+7| 9

0 | -2(0)+7 | 7

1 | -2(1)+7 | 5

A,B, and C are real numbers, A and B cannot both be 0 at the same time, and A must be positive. The exponents of x and y are ones. The graph of these equations always plots in a straight path.

Ex:

Linear=4x+3y=5

Nonlinear=4x2+3y=5

y=n always plots as a horizontal line

y=3

x=n always plots as a vertical line

x=4

Graphing a linear equation by plotting ordered pair solutions. Use a function table to derive ordered pairs

Ex:

y=-2x+7

x | -2x+7 | y

-1 | -2(-1)+7| 9

0 | -2(0)+7 | 7

1 | -2(1)+7 | 5