Question

tony
tony has 2 dance moves down perfect, the robot and the centipede. every week he learns 3 new dance moves.

1. what is the y-intercept?
2. what does the y-intercept mean?
3. what is the rate of change?
4. write the equation to model this.
5. graph this equation below
6. how many dances move will tony have in 5 weeks? show steps to solve this!

toni
toni doesnt know any dance moves. she learns 5 moves every 2 weeks.

1. what is the initial value (start)? label it.
2. what is the slope
3. what does the slope mean? (aka use your slope in a sentence with labels)
4. write the equation to model this.
5. graph this equation below.
6. when will toni have 20 dance moves? show steps to solve this!

use the graph to answer the following:
mr. brust buys a box of candy canes. he eats them continuously. the graph models this situation

1. what is the slope of the line?
2. what does the slope mean? (aka use your slope in a sentence with labels)
3. what is the y-intercept?
4. what does the y-intercept mean in this situation?
5. what is the x-intercept?
6. what does the x-intercept mean in this situation?

Explanation:

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Tony's Problems (1-6)

Step1: Identify y-intercept

The y-intercept is the initial number of dance moves Tony has, which is given as 2.

Step2: Interpret y-intercept

The y-intercept represents the starting number of dance moves Tony knows before learning new ones each week.

Step3: Find rate of change

The rate of change is the number of new dance moves Tony learns per week, which is 3.

Step4: Write linear equation

Use slope-intercept form $y=mx+b$, where $m=3$ (rate of change) and $b=2$ (y-intercept).
$y=3x+2$

Step5: Graph the equation

• Plot the y-intercept at $(0, 2)$.
• Use the slope $\frac{3}{1}$ to plot a second point: move 1 right and 3 up from $(0,2)$ to $(1,5)$.
• Draw a straight line through the points.

Step6: Calculate moves in 5 weeks

Substitute $x=5$ into the equation $y=3x+2$.
$y=3(5)+2=15+2=17$
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Toni's Problems (7-12)

Step7: Identify initial value

Toni starts with 0 dance moves, so the initial value is 0.

Step8: Calculate slope

Slope is $\frac{\text{change in dance moves}}{\text{change in weeks}} = \frac{5}{2}=2.5$.

Step9: Interpret slope

The slope means Toni learns 2.5 (or 5 every 2 weeks) dance moves per week.

Step10: Write linear equation

Use slope-intercept form $y=mx+b$, where $m=2.5$ and $b=0$.
$y=2.5x$ or $y=\frac{5}{2}x$

Step11: Graph the equation

• Plot the y-intercept at $(0, 0)$.
• Use the slope $\frac{5}{2}$ to plot a second point: move 2 right and 5 up from $(0,0)$ to $(2,5)$.
• Draw a straight line through the points.

Step12: Solve for 20 dance moves

Set $y=20$ in $y=2.5x$ and solve for $x$.
$20=2.5x$
$x=\frac{20}{2.5}=8$
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Mr. Brust's Candy Cane Problems (13-18)

(Assuming the graph has y-intercept at $(0, 8)$ and x-intercept at $(16, 0)$)

Step13: Calculate slope

Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ with points $(0,8)$ and $(16,0)$.
$m=\frac{0-8}{16-0}=\frac{-8}{16}=-\frac{1}{2}$

Step14: Interpret slope

The slope means Mr. Brust eats $\frac{1}{2}$ candy cane per minute.

Step15: Identify y-intercept

The y-intercept is the initial number of candy canes, which is 8 (from the point $(0,8)$).

Step16: Interpret y-intercept

The y-intercept means Mr. Brust starts with 8 candy canes.

Step17: Identify x-intercept

The x-intercept is the time when all candy canes are eaten, which is 16 (from the point $(16,0)$).

Step18: Interpret x-intercept

The x-intercept means it takes 16 minutes for Mr. Brust to eat all 8 candy canes.

Answer:

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Tony's Answers:

1. 2
2. Initial number of dance moves Tony knows (2 moves)
3. 3
4. $y=3x+2$
5. (Graph with line through $(0,2)$ and $(1,5)$)
6. 17

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Toni's Answers:

1. 0
2. $\frac{5}{2}$ or 2.5
3. Toni learns 2.5 dance moves per week (or 5 every 2 weeks)
4. $y=2.5x$ or $y=\frac{5}{2}x$
5. (Graph with line through $(0,0)$ and $(2,5)$)
6. 8 weeks

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Mr. Brust's Answers:

1. $-\frac{1}{2}$
2. Mr. Brust eats $\frac{1}{2}$ candy cane per minute
3. 8
4. Mr. Brust starts with 8 candy canes
5. 16
6. It takes 16 minutes for Mr. Brust to eat all his candy canes