there are several ways to graph a linear

Unit 3 Linear Functions pt 1.notebook

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October 21, 2012

Oct 59:37 PM

Graphs, Linear Equations and

Functions

Oct 59:38 PM

There are several ways to graph a linear equation: • Make a table of values• Use slope and y-intercept• Use x and y intercepts

Oct 59:43 PM

Example: Make a table of values and graph 2x+y=5

x y012

Oct 64:20 PM

1 Find the missing value in the table for 2x y = 10.

A 7

B 2

C 7

D 2

x y1 125 04 ?

Oct 64:38 PM

2 Find the value of y when x = 6

Oct 64:42 PM

3 Point A lies on the line x + 3y = 7.

Find the missing coordinate of point A( , 3).


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October 21, 2012

Oct 64:51 PM

4 True or False: The line 2x 6y = 6 goes through the origin.

True

False

Oct 59:55 PM

Using x and y Intercepts Example: Graph 3x+2y=6.

• Find the value of x when y=0. This is where the line crosses the x-axis, and is called the x-intercept. Substituting 0 for y, the equation becomes 3x=6, or x=2. The coordinates of the x-intercept are (2, 0).

• Now find the value of y when x=0. This is where the line crosses the y-axis, called the y-intercept. Substituting 0 for x, the equation becomes 2y=6, or y=3. The coordinates of the y-intercept are (0,3).

Oct 510:03 PM

Graph 3x+2y=6Using the two intercepts, (2,0) and (0,3), we then graph the line.

Oct 64:20 PM

Remember:The x-intercept is where the line crosses the x-axis. The y-coordinate is zero on the x axis.

The y-intercept is where the line crosses the y-axis. The x-coordinate is zero on the y-axis.

y=0

x=0

x=0

Oct 65:03 PM

Vertical lines are parallel to the y-axis. These lines have no y-intercept and have an equation in the form x=k, where k is a constant.For example, the vertical line x=2 is parallel to the y-axis and has an x-intercept of (2,0).

Oct 65:41 PM

Horizontal lines are parallel to the x-axis, so they have no x-intercept. The equations of horizontal lines are in the form y=k, where k is a constant. For example, y=-5 is parallel to the x-axis and has a y-intercept of (0,-5).


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October 21, 2012

Oct 65:51 PM

5 What is the yintercept of ?

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6 What is the xintercept of

?

Oct 65:56 PM

7 Which is the equation of a horizontal line?

A

B

C

D

Oct 66:01 PM

8 Which is the graph of

A B

C D

?

Oct 66:10 PM

9 Which line goes through the origin?

A

B

C

D

Oct 66:15 PM

Slope-Intercept Form The equation for a line may be written as y = mx + b, where m is the slope of the line and b is the y-intercept. For example, the line y=3x+1 has a slope of 3 and a y-intercept of (0,1).Recall that the slope of a line is found by measuring . A line which climbs from left

to right has a positive slope and a line which falls from left to right has a negative slope.


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October 21, 2012

Oct 66:24 PM

10 What is the slope of this line?

Oct 66:29 PM


Oct 66:33 PM


Oct 67:38 PM

Graphing Linear Inequalities

Linear inequalities are shaded regions of the coordinate plane with boundaries that are straight lines When we need to graph an inequality such as 2x+3y≤6, we see that the boundary line has the equation 2x+3y=6. Once we graph the boundary line, we find the region to be shaded by testing a point on either side of the line. If the point makes the inequality true, it is part of the region to be shaded. If it does not make the inequality true, then the region on the opposite side of the boundary line must be shaded.

Oct 67:52 PM

Graphing 2x+3y≤6: Graph the boundary line 2x+3y=6. Since the inequality allows it to be equal to 6, we use a solid line. (Otherwise, the line is dotted). The x-intercept is 3 and the y-intercept is 2. So, the line passes through (3,0) and (0,2)Now test a point on one side of the line. We can use (0,0) as long as the line does not pass through the origin.Is 2(0)+3(0) ≤ 6? Yes, 0≤6. So, we shade the side of the line that contains (0,0).

Oct 68:02 PM

Try this one: 4x-y≥8


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October 21, 2012

Oct 68:04 PM

Does your graph look like this?

Oct 66:34 PM

Finding the Slope of a LineWhen the equation of the line is in the form Ax+By=C where A, B and C are integers and A is positive, it is in standard form. We can find the slope of the line in standard form by solving the equation for y.Example: 5x-2y=8

-2y=-5x+8y=(5/2)x-4

From y=mx+b, we can see that the slope is 5/2 and the y-intercept is (0,-4).

Subtract 5x from both sides

Divide both sides by -2

Oct 66:55 PM

13 Find the slope of the line

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14 What is the yintercept of the line

?

Oct 66:59 PM

Finding the slope from 2 points on the line:

The slope formula can be used to

find the slope of a line from the coordinates of any two points on the line. Example: The slope of the line which goes through (1,3) and (5,9) is equal to

Oct 67:11 PM

15 Find the slope of the line which goes through (3,1) and (2, 4).


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October 21, 2012

Oct 67:03 PM

Note! It is impossible to find the slope of a vertical line because the value of x is constant on the line. We say that the slope of a vertical line is undefined. If we try to use the slope formula, the denominator becomes zero, and it is impossible to divide by zero.

Oct 67:17 PM

Point-Slope Form:

This form of an equation for a straight line is based on the fact that the slope of the straight line is the same between any two points on the line. Using point-slope form is an easy way to find the equation for a line when we know the slope, m, and one of the points it passes through (x1,y1).

Oct 67:24 PM

Example: Let's find an equation for the line that has a slope of 2 and passes through (3, 5).

Using point-slope form,

Substitute the values for m, x1 and y1.

Use the distributive property.

Add 5 to both sides.

Now that the equation is in slope-intercept form, we can see that the y-intercept is (0,-1).

Oct 69:12 PM

Equation Description When to Use

Slope-intercept form. Slope is m, y-intercept is (0,b).

Graphing the equation using the y-intercept and rise/run from the slope.

Point slope form. Slope is m and the line passes through (x1, y1)

Finding the equation of a line when the slope and one or two points on the line are known

Standard Form. A, B and C are integers and A is positive. Slope is -A/B, B≠0. x-intercept is (C/A, 0), C≠0, y-intercept is (0,C/B), B≠0.

The x and y intercepts can be found quickly and used to graph the equation without having to calculate the slope.

Horizontal Line. Slope is 0, and y-intercept is (0,b).

If the graph intersects only the y axis, then y is the only variable in the equation.

Vertical Line. Slope is undefined. x-intercept is (a, 0).

If the graph intersects only the x axis, then x is the only variable in the equation.

Summary: Forms of Linear Equations

Oct 72:30 PM

Writing Equations for Lines

If we know the slope of a line and the y-intercept, we can just substitute into slope-intercept form.Example: Write an equation for the line with slolpe of -2 and y-intercept of (0,7). Using y=mx+b, we get y=-2x+7 for the line.

Oct 72:34 PM


If we know the slope of the line and a point is passes through, we can use point-slope form.Example: Find an equation for the line that passes through the point (3, 7) as is parallel to y=3x-1.We know the slope has to be 3, because parallel lines have the same slope.Using point-slope form, we get


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October 21, 2012

Oct 72:41 PM


Write an equation for a line that is perpendicular to y=2x-3 and passes through the point (1,5).The slope of the line has to be the negative reciprocal of 2 to make it perpendicular to a line with slope 2.Again,using point-slope form, we get

Oct 72:47 PM

Writing Equations for LinesIf we need to find an equation for the line parallel to a line in standard form, there is an easier way. If you notice, parallel lines when written in standard form have the same coefficients for x and y.Example: Find an equation for the line parallel to 2x+3y=9 that passes through (-3, 4).We know the line will be 2x+3y=C, and we can substitute for x and y from the point to find C.2(-3)+3(4) = -6+12=6, so C=6, and the equation of the line is 2x+3y=6

Oct 72:54 PM

Writing Equations for LinesIf we need to find an equation for the line perpendicular to a line in standard form, there is an easier way. If you notice, perpendicular lines when written in standard form have the coefficients for x and y switched and the opposite sign between the two terms.Example: Find an equation for the line perpendicular to 2x+3y=9 that passes through (1,3).We know the line will be 3x-2y=C, and we can substitute for x and y from the point to find C.3(1)-2(3) = 3-6=-3, so C=-3, and the equation of the line is 3x-2y=-3

Oct 85:09 PM

Write an equation for the line parallel to y=3x1 which goes through the point (2,4).

Oct 85:10 PM

Write an equation for the vertical line which passes through (3,2).

Oct 85:11 PM

Write an equation for the line which is perpendicular to 3x4y=16 and passes through (3,4).


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October 21, 2012

Oct 85:13 PM

Write an equation for the line which is perpendicular to y=1/2x+3 and goes through (0,1).

Oct 85:14 PM

Write an equation for the line with a slope of 5 which passes through (0,3).

Oct 85:15 PM

Write an equation for the line with a slope of 4 which passes through the point (1/2, 3).

Oct 85:15 PM

The table below shows the coordinates of points on a line. What is an equation for the line?

x y5 77 1110 17

Oct 69:14 PM

Applications of Linear EquationsSuppose you are filling your car with gasoline. The price at the local gas station is $3.60 per gallon. Write an equation that describes the cost y to buy x gallons of gas. As you pump the gas, two numbers spin by, the number of gallons purchased and the price. The table summarizes the ordered pairs (x,y)

Number of Gallons Purchased=x

Price of the Number of Gallons=y

0 0(3.60)=$0

1 1(3.60)=$3.60

2 2(3.60)=$7.20

3 3(3.60)=$10.80

4 4(3.60)=$14.40

The equation y = 3.6x models this relationship, since we multiply the number of gallons by $3.60 to get the price.

Oct 710:50 AM

Suppose the same gas station was offering a car wash for $3.00 with every fill up. Write an equation to find the price for gas and a car wash.

Since an additional $3.00 be charged, the equation becomes y= 3.6 x + 3

What would the price be for 7 gallons of gas and a car wash?

$28.20


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October 21, 2012

Oct 710:58 AM

Linear ModelsSometimes data collected over a period time can be modelled by an equation for a straight line. The table below shows the average annual tuition and fees at four year colleges over a 12 year period.

Year Cost (in Dollars)1990 2035

1994 2820

1996 3151

1998 3486

2000 3774

2002 4273

Oct 711:22 AM

Linear ModelsWe first define x as the number of years since 1990. So in 1990, x=0, in 1994, x=4, etc. Then we can find the slope using the 2 end points of the data, (0, 2035) and (12,4273).

Year x Cost in dollars = y

1990 0 2035

1994 4 2820

1996 6 3151

1998 8 3486

2000 10 3774

2002 12 4273

Use slopeintercept form

Use the slope formula

Oct 711:25 AM

Now that we have found an equation to model the data, y=186.5x+2035, we can use the equation to predict what the cost of tuition and fees will be in 2016.

Since x represents the number of years since 1990, substitute x=16 into the linear equation.

We predict the cost of tuition and fees to be $5019 in 2014.

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Oct 711:48 AM

Write a linear equation to model the data in the table.

Year Travel Expenses

2001 5002002 7502003 10002004 1250

there are several ways to graph a linear

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