The Cartesian Coordinate Plane

7th Grade Math A Handwritten QuizThis Presentation is from the book:

Math For Everyone™Copyright © 2007 Nathaniel Max Rock

MathForEveryone.com

7th Grade Math A Handwritten Quiz Prompt:No Abbreviations, Show All Your Work, No Calculators

1. What is the y-intercept?2. What is the rate or slope?3. Write an equation to represent this situation.4. What is the independent variable? What is the dependent

variable?5. What is the x-intercept?6. At what time period is the “break even point” reached?7. Given a new option, what is the y-intercept of the new option?8. What is the rate or slope of the new option?9. Write an equation to represent the new option.10. What is the x-intercept of the new option?11. At what time period are the two options equal? At what value are

the two options equal?12. Graph and label the two options on the same graph. Illustrate the

slopes of the lines with labeled rise and run lines.13. Bonus: Which option would you choose and why?

Given:

A business costs $500 of investment to start

$100 per week can be generated

1. What is the y-intercept?

In graphing, the y-intercept is the starting point At the y-intercept the value of x is zero The coordinate at the y-intercept is (0,y) In business, investment is first, profit is later Investment is therefore a negative number Investment is the starting point In this case, a $500 investment is required Therefore we have -500 Or a coordinate of (0,-500)

Starting Point, y-intercept

x

y

Investment is negative (the opposite of profit)

The y-intercept = -500Or Coordinate (0,-500)

(0,-500)

2. What is the Slope or Rate?

This business generates $100 per week This is one thing in relation to another Money ($) per time (week) $/week $100 is made for every week which

passes

“Rise” over “Run”

x

y

Rise =

x1

y1 (x1,y1)

x2

y2(x2,y2)

y2-y1

Run = x2-x1

Slope =Rise = y2-y1

Run = x2-x1

Slope or Rate, Continued

run

riseSlope

2 1

2 1

( )

( )

y y

x x

=m

Slope =m

2 1

2 1

( )

( )

y ym

x x

Memorize this!

Memorize this!

Slope or Rate, Continued

2 1

2 1

( )

( )

y ym

x x

2 1

2 1

( )

( )

y ym

x x

2 1( )

1

x x2 1( )

1

x x

2 1 2 1( ) ( )x x m y y

2 1 2 1( ) ( )y y m x x

* *( )y y m x x Memorize this!Point-slope form

Slope or Rate, Continued

* *( )y y m x x At the y-intercept, the value of x is zero! (0,y)

* ( (0))y y m x

*y y mx

* * *y y y mx y

*y mx y

y mx b

*b y

Memorize this!Slope-intercept form

Standard & Slope-Intercept Form

The standard form ax+by=c And the slope intercept form y=mx+b Are related but the b’s are different! Let’s have a look:

ax by c ax ax

by ax c

Standard & Slope-Intercept Form Continued by ax c

b bax c

yb b

a cy x

b b

standardform

am

b

y-interceptstandardform

cy - intercept b

b

y-intercepty mx b

Standard Formax by c

aslope m

b

(0, )y - intercept y

(0)a by c ( ,0)x - intercept x

(0)ax b c ax cax c

a a

interceptc

xa

ax by c

by cby c

b b

interceptc

yb

Memorize these!

Back to the problem…

The initial investment of $500 is negative The coordinate is (0,-500) The rate or slope is $100 per week Slope is rise over run so 100/1 Slope is always from out starting point Our starting point is (0,-500) From this point rise 100 and run 1

Back to the problem…

x

y

From the starting point,Rise 100, Run 1

(0,-500)

3. The Equation of the Line

The line has an equation The y-intercept is -500 Our slope is m=100/1=100 The slope-intercept form is y=mx+b (b is the y-intercept) Therefore: y=100x-500 This is the slope-intercept form equation of this

line Label the line

4. Independent, Dependent?

The amount of money depends on the weeks The number of weeks are independent of money The x-axis is always the independent variable The y-axis is always the dependent variable The y value dependents on the x value

5. What is the x-intercept?

x

y

y=100x-500

(0,-500)

What is the value of the x-

intercept?

The x-intercept Continued

Where the graph crosses the x-axis Visually we can see the position at (5,0) How can we calculate this value? The value of y=0 at the x-intercept or (x,0) Therefore, in y=100x-500, We now write 0=100x-500 Now solve for x

The x-intercept Continued

0 100 500x 0 500 100 500 500x

500 100x100 500x 100 500

100 100

x

5x x-intercept coordinate (5,0)

Back to the problem…

x

y

y=100x-500

(0,-500)

(5,0)

Labeling the Axes

The x variable is the independent The y variable is dependent Therefore x represents weeks Therefore y represents money, or $ The money values are large, therefore Each y interval represents $100 The x-axis is weeks The y-axis is money in $100 intervals

Back to the problem…

x

y

y=100x-500

(0,-500)

(5,0)(weeks)

($100’s of dollars)

6.The Break Even Point

Not negative Not positive In-between—Zero The break even point is the zero point Where the value of y is zero Therefore: 0=100x-500, solve for x Just like for the x-intercept! Same process, same value, but different use

The Break Even Point Continued0 100 500x

0 500 100 500 500x 500 100x

100 500x 100 500

100 100

x

5x It will take 5 weeks to earn back the initial

investment of $500

7. Given the Option…

There is an option To buy a more expensive business for

$700 Which generates more money per week—

$200 The y-intercept is the starting point or -700 The coordinate of this point is (0,-700)

8. What is the Slope or Rate?

The slope or rate is $200 per week Rise over run is 200 over 1 or 200/1

9. Equation of the Option?

The y-intercept is -700 The slope or m is 200 Fill in the slope-intercept equation y=mx+b Therefore: y=200x-700

10. The x-intercept of the Option

0 200 700x 0 700 200 700 700x

700 200x200 700x 200 700

200 200

x

3.5x x-intercept coordinate (3.5,0)

11. When Will The Two Options Be Equal?

100 500 200 700x x

2x

100 500 500 200 700 500x x 100 200 200x x

100 200 200 200 200x x x x ( 100 200)( 1)x

100 200x 100 200

100 100

x

The two options are equal at 2 weeks.

11. Continued, What will be the Value of the Function When the Two Options are Equal?

100 500y x

The two options are equal when $300 is still owed.

Choose either equation and plug in the value of x=2

100(2) 500y 200 500y 300y

Bonus: The “Solution”

(2, 300)The solution to the two equations is

Graph & Label

x

y

y=100x-500

(0,-500)

(5,0)(weeks)

($100’s of dollars)

(0,-700)

(3.5,0)

y=200x-700

(2,-300)

positive slopem=+200/1

rise 200, run 1

positive slopem=+100/1

rise 100, run 1

the “solution” to the two equations

x-intercepts

y-intercepts