The Cartesian Coordinate Plane 7 th Grade Math A Handwritten Quiz This Presentation is from the book: Math For Everyone™ Copyright © 2007 Nathaniel Max Rock MathForEveryone.com

7 th 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 = -500 Or 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 = x1x1 y1y1 (x 1,y 1 ) x2x2 y2y2 (x 2,y 2 ) y 2 -y 1 Run = x 2 -x 1 Slope = Rise = y 2 -y 1 Run = x 2 -x 1

Slope or Rate, Continued Slope =m=m =m=m Memorize this!

Slope or Rate, Continued Memorize this! Point-slope form

Slope or Rate, Continued At the y-intercept, the value of x is zero! (0,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:

Standard & Slope-Intercept Form Continued

Standard Form 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 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 Continued 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 x-intercept coordinate (3.5,0)

11. When Will The Two Options Be Equal? The two options are equal at 2 weeks.

11. Continued, What will be the Value of the Function When the Two Options are Equal? The two options are equal when $300 is still owed. Choose either equation and plug in the value of x=2

Bonus: The “Solution” 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 slope m=+200/1 rise 200, run 1 positive slope m=+100/1 rise 100, run 1 the “solution” to the two equations x-intercepts y-intercepts