Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find...

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Chapter 4 Reivew Chapter 4 Reivew Graphing Linear Equations Graphing Linear Equations

Transcript of Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find...

Page 1: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

Chapter 4 ReivewChapter 4 ReivewChapter 4 ReivewChapter 4 Reivew

Graphing Linear EquationsGraphing Linear Equations

Page 2: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 3: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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.

Page 4: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

#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

Page 5: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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.

Page 6: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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)

Page 7: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 8: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 9: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 10: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 11: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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.

Page 12: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 13: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 14: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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)

Page 15: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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)

Page 16: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 17: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 18: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 19: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 20: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 21: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

PositiveNegative

Zero

Undefined

1. 2.

3. 4.

Page 22: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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.

Page 23: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

Find the slope of each line?

m = 4/5m = -1/2

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

Page 24: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

Find the slope of each line below:

Line AE =

Line DB =

Line CB =

-5/6

3/7

2

Page 25: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 26: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

slope 2.

Page 27: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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??

Page 28: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 29: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 30: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 31: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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)

Page 32: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 33: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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??

Page 34: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 35: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

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

Page 36: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 37: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 38: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 39: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 40: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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)

Page 41: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 42: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 43: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 44: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

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

Page 45: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

y = 2 y = #

Horizontal Line

Page 46: Chapter 4 Reivew Graphing Linear Equations. LIST OF TOPICS ON THIS TEST: 1. Plotting points 2. Find the x-intercept 3. Find the y-intercept 4. Write the.

x = −3 x = #

Vertical Line