Chapter 4 ReivewChapter 4 ReivewChapter 4 ReivewChapter 4 Reivew

Graphing Linear EquationsGraphing Linear Equations

LIST OF TOPICS ON THIS TEST:

1. Plotting points2. Find the x-intercept3. Find the y-intercept4. Write the equation given in slope-intercept form5. Find the slope between two points6. Look at a graph of a line and find the slope7. Does this table represent a function?8. Evaluate a function9. Are these lines parallel?10. Graph 6-8 lines

A linear equation is an equation whose solutions fall on a STRAIGHT line on the coordinate plane.

All solutions of the equation are on this line.

Look at the graph to the left, points (1, 3) and (-3, -5) are found on the line and are solutions to the equation.

#2 and #3 #2 and #3 x-intercepts and x-intercepts and

y-interceptsy-intercepts

#2 and #3 #2 and #3 x-intercepts and x-intercepts and

y-interceptsy-interceptsPlug in zero for the oppositePlug in zero for the opposite

x-intercept – the x-coordinate of the point where the graph of a line crosses the x-axis (where y = 0).

y-intercept – the y-coordinate of the point where the graph of a line crosses the y-axis (where x = 0).

Slope-intercept form (of an equation) – a linear equation written in the form y = mx +b, where m represents slope and b represents the y-intercept.

To find the “x” and “y” intercept of the graph, you substitute 0 for y and solve for x. then substitute 0 for x and solve for y.

2x + 3y = 62x + 3(0) = 6 2x = 6 x = 3

The x-intercept is 3.(3,0)

2x + 3y = 62(0) + 3y = 6 3y = 6 y = 2

The y-intercept is 2. (0,2)

Find the x-intercept and y-intercept of each line. Use the intercepts to graph the

equation.

1) x – y = 5

2) 2x + 3y = 12

3) 4x = 12 + 3y

4) 2x + y = 7

5) 2y = 20 – 4x

(0,-5) and (5,0)

(0,4) and (6,0)

(0,-4) and (3,0)

(0,7) and (7/2,0)

(0,10) and (5,0)

4x – 3y = 12

4x + 2y = 20

4. Write the equation 4. Write the equation in slope-intercept in slope-intercept

formform

4. Write the equation 4. Write the equation in slope-intercept in slope-intercept

formform

y = mx + by = mx + b

Slope-intercept Form

• An equation whose graph is a straight line is a linear equation. Since a function rule is an equation, a function can also be linear.

• m = slope• b = y-intercept

Y = mx + b

For example in the equation;y = 3x + 6

m = 3, so the slope is 3b = +6, so the y-intercept is (0,6)

Let’s look at another:y = 4/5x -7

m = 4/5, so the slope is 4/5b = -7, so the y-intercept is (0,-7)

Please note that in the slope-intercept formula;y = mx + b

the “y” term is all by itself on the left side of the equation.

That is very important! YOU MUST SOLVE FOR Y FIRST

y=2x+1

y=−x−4

y=32

x−2

These equations are all in Slope-Intercept Form:

Notice that these equations are all solved for y.

Just by looking at an equation in this form, we can see the slope and y-int. •The constant is the y-intercept.

•The coefficient is the slope.

y=2x+1

y=−x−4

y=32

x−2

y-intercept = (0,1)

slope = 2/1

y-intercept = (0, -4)

slope = -1/1

y-intercept = (0,-2)

slope = 3/2

Sometimes we must solve the equation for y before we can graph it.

2x+y =3

y=−2x+3

b = 3 is the y-intercept. (0,3)

The coefficient, m = -2/1 is the slope.

-2x -2x

Write the slope-intercept form of the equation to determine the

slope and y-intercept.

3x – y = 14-3x -3x-y = -3x + 14-1 -1 -1y = 3x – 14 m = 3/1b = (0,-14)

x + 2y = 8-x -x2y = -x + 8 2 2 2y = -1x + 4 2

m = -1/2b = (0,4)

Write each equation in slope-intercept form.

Identify the slope and y-intercept.

1.) 2x + y = 10 2.) -4x + y = 6

3.) 4x + 3y = 9 4.) 2x + y = 3

5.) 5y = 3x

-4x -4x3y = -4x + 9

3 3 3

€

y =−4

3x + 3

m = -4/3b = (0,3)

-2x -2x

y = -2x + 10m = -2/1b = (0,10)

4x 4xy = 4x + 6

m = 4/1b = (0,6)

-2x -2xy = -2x + 3 m = -2/1

b = (0,3)

5 5

€

y =3

5x

m = 3/5b = (0,0)

5. Finding Slope 5. Finding Slope between two points between two points

given.given.

5. Finding Slope 5. Finding Slope between two points between two points

given.given.SLOPE FORMULASLOPE FORMULA

Find the slope of the line that passes through each pair of points.

1) (1, 3) and (2, 4)

2) (0, 0) and (6, -3)

3) (2, -5) and (1, -2)

4) (3, 1) and (0, 3)

5) (-2, -8) and (1, 4)

€

m =4 − 3

2 −1=

1

1=1

€

m =−3 − 0

6 − 0=

−3

6=

−1

2

€

m =−2 − −5

1 − 2=

3

−1= −3

€

m =3 −1

0 − 3=

2

−3=

−2

3

€

m =4 − −8

1− −2=

12

3= 4

Find the slope of the line that passes through each pair of points.

6) (1, 3) and (-1, -1)

7) (2,3) and (2,-3)

8) (3,1) and (6,0)

9) (4,4) and (2,4)

€

m =−1− 3

−1 −1=

−4

−2= 2

€

m =−3 − 3

2 − 2=

−6

0= UNDEFINED

€

m =0 −1

6 − 3=

−1

3

€

m =4 − 4

2 − 4=

0

−2= 0

6. Finding Slope by 6. Finding Slope by looking at a graphlooking at a graph

6. Finding Slope by 6. Finding Slope by looking at a graphlooking at a graph

Count your boxesCount your boxes

Find the slope by looking at a graph:

Just count your boxes from one point to another.

Slope = rise run

Up 2Right 3

€

m =2

3

Slopes: positive, negative, no slope (zero), undefined.

PositiveNegative

Zero

Undefined

1. 2.

3. 4.

First tell me if the slope is positive or negative, then find the slope of each line,

1. Positive

m = 2/3

2. Negative

m = -2

3. Zero Undefined4.

Find the slope of each line?

m = 4/5m = -1/2

First find two points to use that cross the grid nicely.

Find the slope of each line below:

Line AE =

Line DB =

Line CB =

-5/6

3/7

2

Use the graph to find the slope of the line.

Sometimes, there will not be points given on the line, you will have to choose nice points that cross the grid nicely.

m = -2

Graphing a Line Using a Point and the SlopeGraph the line passing through (1, 3) with

slope 2.

7. Does this 7. Does this table/graph represent table/graph represent

a function??a function??

7. Does this 7. Does this table/graph represent table/graph represent

a function??a function??

Does this table represent a function?

Remember, each input MUST go to only one output!!!

No because the 3 goes to 1 and -1

No because the 2 goes to B and C

YES

Ex 1: Determine whether each relation is a function.

a. {(4,5), (6,7), (8,8)}

b. {(5,6), (4,7), (6,6), (6,7)}

Yes!!!!

NO!!!!

Because the 6 outputs a 6 and a 7

Your Turn – Identifying a Function

• Does the table represent a function? Explain

Input Output

1 3

1 4

2 5

3 6

Input Output

1 1

2 3

3 6

4 10

Input Output

1 3

2 6

3 11

4 18

Input Output

5 9

4 8

3 9

2 7

1.

2.

3.

4.

YESNo

YES

YES

8. Review how to 8. Review how to evaluate a functionevaluate a function8. Review how to 8. Review how to

evaluate a functionevaluate a function

f(x)f(x)

Ex 3: Evaluating a Function

If f(x) = 3x – 2, evaluate the function when:a. x = -1 b. x = 2 c. x = -4

3(1) - 2

3 - 2 = 1 6 – 2 = 4 -12 – 2 = -14

3(2) - 2 3(-4) - 2

9. Are these two lines 9. Are these two lines parallel??parallel??

9. Are these two lines 9. Are these two lines parallel??parallel??

Do they have the Do they have the same slope??same slope??

Are these lines parallel?

1.) y = 2x -1 y = -2x + 1 NO

m = 2

m = -2

2.) y = 4x - 7 NO

m = 1/4

m = 4

3.) 5x – y = 4 -5x + y = 15

YES m = 5

m = 5

4.) 3x – 2y = 6 2y = -3x + 12

NO

m = 3/2

m = -3/2

€

y =1

4x + 5

***make sure you put equation into slope-int first***

y = 5x - 4

y = 5x + 15

€

€

y =3

2x − 3

€

y =−3

2x + 6

10. Graph Using 10. Graph Using Slope-Intercept FormSlope-Intercept Form

10. Graph Using 10. Graph Using Slope-Intercept FormSlope-Intercept Form

y=2x+1

1) Plot the y-intercept as a point on the y-axis, the y-intercept = (0,1)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The slope = 2/1

up 2

right 1 up 2

right 1

NOW LET’S GRAPHUSING SLOPE-INT.FORM

1) Plot the y-intercept as a point on the y-axis, the y-intercept = (0,-4)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -1, so the slope = -1/1

right 1down 1

right 1

y=−x−4

down 1

1) Plot the y-intercept as a point on the y-axis. The constant, b = -2, so the y-intercept = (0,-2)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 3/2, so the slope = 3/2

right 2

up 3

y=32

x−2

right 2

up 3

1) Plot the y-intercept as a point on the y-axis. The constant, b = 3, so the y-intercept = (0,3)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -2, so the slope = -2/1

right 1down 2

right 1

down 2

y=−2x+3

Now let’s use slope-intercept form to graph

6x – 3y = 12

You must rewrite the equation 6x – 3y = 12 in slope-intercept to be able to identify the slope and

y-intercept.

6x – 3y = 12-6x -6x

– 3y = -6x + 12–3 -3 -3

y = 2x - 4

Steps to graphing using slope-int.1.Graph the y-int.2.Use the slope to graph other points on the line

m = 2

b = (0,-4)

Graph the equationy = 2x + 6

by using slope-intercept form

Step 1 – Graph the y-intercept

b = (0,6)

Step 2 – use the slope to find the other points

m = 2/1

1) Plot the y-intercept as a point on the y-axis. The constant, b = -1, so the y-intercept = (0,-1)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 2/3, so the slope = 2/3

€

3y = 2x − 3 Solve for “y” first

€

y =2

3x −1

1) Plot the y-intercept as a point on the y-axis. The constant, b = 0, so the y-intercept = (0,0)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 1/3, so the slope = 1/3

€

3y = x Solve for “y” first

€

y =1

3x

1) Plot the y-intercept as a point on the y-axis. The constant, b = 3, so the y-intercept = (0,3)

2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -1/4, so the slope = -1/4

€

x + 4y =12 Solve for “y” first

€

y =−1

4x + 3

€

y = 2 y = #

Horizontal Line

€

x = −3 x = #

Vertical Line