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Ryan is tracking how much money he has each week.

He started with $50.
He saves $15 per week for the first 6 weeks.
He spent $10 per week after week 6 until the money's all spent.
(4 points)
1) The graph of this scenario would be (response area) because the rate of change changes.
responses
linear
nonlinear
2) The y-intercept of this graph would be (responses area) meaning (responses area)
1st responses
(0, 50)
(0, 5)
(0, 10

2nd responses
Ryan started at 6 weeks
Ryan started with 10$
Ryan started with $50

3) The rate of change is (response area) after the first 6 weeks because as the weeks increase the money decreases.
responses
negative
positive

11 answers

  1. The graph of this scenario would be nonlinear because the rate of change changes.

  2. The y-intercept of this graph would be (0, 50) meaning Ryan started with $50.

  3. The rate of change is negative after the first 6 weeks because as the weeks increase, the money decreases.

The equation 10x+3y=90
is in standard form. This form tells us the x and y-intercepts easily.

What are the x and y-intercepts?

(1 point)
Responses

The x-intercept is (9,0) and the y-intercept is at (0,30).
The x-intercept is (9,0) and the y-intercept is at (0,30).

The x-intercept is (10,0) and the y-intercept is at (0,3).
The x-intercept is (10,0) and the y-intercept is at (0,3).

The x-intercept is (1/9,0) and the y-intercept is at (0,1/30).
The x-intercept is (1/9,0) and the y-intercept is at (0,1/30).

The x-intercept is (3,0) and the y-intercept is at (0,10).

To find the x and y-intercepts of the equation \(10x + 3y = 90\):

  1. Finding the x-intercept:

    • To find the x-intercept, set \(y = 0\): \[ 10x + 3(0) = 90 \implies 10x = 90 \implies x = 9 \] So, the x-intercept is \((9, 0)\).

  2. Finding the y-intercept:

    • To find the y-intercept, set \(x = 0\): \[ 10(0) + 3y = 90 \implies 3y = 90 \implies y = 30 \] So, the y-intercept is \((0, 30)\).

Thus, the correct response is:

The x-intercept is (9,0) and the y-intercept is at (0,30).

Michele has $55 after 2 weeks of saving money. She plans to have $105 after 7 weeks of saving.

Which equation matches this scenario?

(1 point)
Responses

10x+7y=55
10 x plus 7 y is equal to 55

y−55=10(x−2)
y minus 55 is equal to 10 times open paren x minus 2 close paren

y=7x+55
y is equal to 7 x plus 55

y+7=10(x+2)

To determine which equation matches Michele's saving scenario, we can analyze the information provided:

  • At 2 weeks, she has $55.
  • At 7 weeks, she plans to have $105.

First, we can find the rate of change and the equation of the line. The change in money from week 2 to week 7 is:

\[ $105 - $55 = $50 \]

The time over which this change occurs is:

\[ 7 \text{ weeks} - 2 \text{ weeks} = 5 \text{ weeks} \]

Now we can calculate the rate of saving per week:

\[ \text{Rate} = \frac{50}{5} = 10 \text{ dollars per week} \]

Now we know that Michele saves $10 per week, starting from $55:

The savings can be expressed using the point-slope form starting from the point (2, 55):

\[ y - 55 = 10(x - 2) \]

Now, let's evaluate the options:

  1. 10x + 7y = 55 - This does not match.
  2. y − 55 = 10(x − 2) - This matches the scenario perfectly.
  3. y = 7x + 55 - This does not match.
  4. y + 7 = 10(x + 2) - This does not match.

Thus, the correct response is:

y − 55 = 10(x − 2).

Using the table of values:
x cups of milk y tbl chocolate syrup
2 1
4 2
5 2.5
30 15
(3 points)
1) What is the slope in context?

2) What is the y-intercept in context?

3) What is the equation of the line?

To analyze the table of values given:

  • \(x\) (cups of milk): [2, 4, 5, 30]
  • \(y\) (tablespoons of chocolate syrup): [1, 2, 2.5, 15]
  1. What is the slope in context?

To find the slope, we take two points from the table. Let's use the points (2, 1) and (4, 2):

\[ \text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{4 - 2} = \frac{1}{2} \]

In context, the slope can be interpreted as follows: For every additional cup of milk, you need 0.5 tablespoons of chocolate syrup.

  1. What is the y-intercept in context?

To find the y-intercept, observe that this is where \(x = 0\). Since the table does not include \(x = 0\), we need to extrapolate. However, we can imply that if we have no cups of milk, we likely wouldn't use any chocolate syrup, suggesting that the y-intercept is at (0, 0).

In context, this means: If there are 0 cups of milk, then 0 tablespoons of chocolate syrup are needed.

  1. What is the equation of the line?

Using the slope we found (\(m = \frac{1}{2}\)) and the y-intercept (0, 0), the equation of the line in slope-intercept form \(y = mx + b\) can be written as:

\[ y = \frac{1}{2}x \]

In context, this means: The number of tablespoons of chocolate syrup (y) is equal to 0.5 times the number of cups of milk (x).

Summary:

  1. The slope in context is 0.5 tablespoons of chocolate syrup per cup of milk.

  2. The y-intercept in context is 0 tablespoons of chocolate syrup when there are 0 cups of milk.

  3. The equation of the line is y = \frac{1}{2}x.

responses
no cups or milk, no syrup
1 tbl syrup to 2 cups of milk
y = 1/2x
y = 2x + 1

Given the options provided for your question, let's match the responses to the context of the scenario:

  1. No cups of milk, no syrup: This is an interpretation of the y-intercept, indicating that if there are 0 cups of milk, there will also be 0 tablespoons of syrup.

  2. 1 tbl syrup to 2 cups of milk: This corresponds to the slope we calculated (1/2), which means for every 2 cups of milk, there is 1 tablespoon of syrup.

  3. y = 1/2x: This is the correct equation derived from the context of the scenario, indicating the relationship between cups of milk (x) and tablespoons of syrup (y).

  4. y = 2x + 1: This equation does not correspond to the data provided and is incorrect based on the slope and y-intercept we've determined.

Summary Selection:

  • For the y-intercept, you can use: No cups of milk, no syrup.
  • For the slope, you could state: 1 tbl syrup to 2 cups of milk.
  • For the equation: y = 1/2x is the correct choice.

So the correct responses in context would be:

  1. No cups of milk, no syrup (for the y-intercept).
  2. 1 tbl syrup to 2 cups of milk (for the slope).
  3. y = 1/2x (for the equation of the line).
The graph below represents the total amount of chips in the bag in oz., y, over time, x, in minutes.
x + 2y = 7

(3 points)
1) What is the end behavior of the graph? Response area
responses
as x goes to -infinity, y goes to + infinity. as x goes to + infinity, y goes to - infinity
as x goes to - infinity, y goes to - infinity. as x goes to + infinity, y goes to infinity
2) What would be a reasonable domain for this scenario? Response area
responses
(0,7)
(0, 3.5)
3) What would be reasonable outputs for the scenario domain? Response area
responses
(0, 3.5)
(0,7)
(0,8)

To analyze the equation given (the equation of the line) and answer the questions properly, we first need to rewrite the equation in slope-intercept form (y = mx + b):

Given the equation: \[ x + 2y = 7 \]

We can solve for \(y\): \[ 2y = 7 - x \] \[ y = \frac{-1}{2}x + \frac{7}{2} \]

1) What is the end behavior of the graph?

The slope of the line is negative (\(-\frac{1}{2}\)), which indicates that the line will decrease as \(x\) increases. The end behavior can be described as follows:

  • As \(x\) goes to \(-\infty\), \(y\) increases toward \(+\infty\).
  • As \(x\) goes to \(+\infty\), \(y\) decreases toward \(-\infty\).

Thus, the correct response is: as x goes to -infinity, y goes to + infinity. as x goes to + infinity, y goes to - infinity.

2) What is a reasonable domain for this scenario?

Given that the context is about the total amount of chips in a bag over time, it doesn't make sense to have negative time or negative amounts of chips.

A reasonable domain based on the scenario might start at \(0\) (no time has passed) and go up to the maximum amount of chips. From the original equation, when \(y = 0\) (0 chips), we can find the corresponding \(x\): \[ x + 2(0) = 7 \implies x = 7 \]

Thus, a reasonable domain could be from (0, 7), where \(0\) representing no time and \(7\) ounces being the maximum chips.

So, the answer is: (0, 7).

3) What would be reasonable outputs for the scenario domain?

The output \(y\) represents the total amount of chips in the bag. Since the total is \(7\) ounces as outlined by the line equation, the outputs would reasonably fall within the range of \(0\) to \(7\) ornaments (the content of the bag).

Response option based on the outputs: The reasonable outputs for the scenario domain would be: (0, 7).

Final Answers Summary:

  1. as x goes to -infinity, y goes to + infinity. as x goes to + infinity, y goes to - infinity.
  2. (0, 7)
  3. (0, 7)