Question

1. Maryann is tracking the change in her vertical jump over 6 months. Use the table to write a linear function that models her jump distance.

Month Vertical Jump in inches
0 16
2 17
4 18
6 19


A.f of x equals one half times x plus 16

B.f of x equals one half times x plus 19

C.f(x) = 2x + 16

D.f(x) = 2x + 19



2. What is the equation of a line that contains the points (5, 0) and (5, −2)?



A.x = 5

B.x = 0

C.y = 0

D.y = 5

3. Choose the equation that represents a line that passes through points (−1, 2) and (3, 1).





A.4x − y = −6

B.x + 4y = 7

C.x − 4y = −9

D.4x + y = 2

4Jewels has $6.75 to ride the ferry around Connecticut. It will cost her $0.45 every time she rides. Identify the dependent variable and independent variable in this scenario.



 A. The number of rides is the independent variable, and the total cost is the dependent variable.

B. The total cost is the independent variable, and the number of rides is the dependent variable.

C. The number of rides and the total cost are both independent variables.

D. The number of rides and the total cost are both dependent variables.

Answer 1

The correct linear function for Maryann's vertical jump is f(x) = ½x + 16. The equation of the line containing points (5, 0) and (5, −2) is x = 5. For the line through points (−1, 2) and (3, 1), the correct equation is 4x − y = −6, and in Jewels' scenario, the number of ferry rides is the independent variable, and the total cost is the dependent variable.

• To write a linear function that models Maryann's vertical jump over 6 months, we need to determine the slope (rate of change) and the y-intercept (starting value).
• Given the increments of 1 inch over every 2 months, we can calculate the slope as 1 inch per 2 months or 0.5 (one half) inches per month.
• The starting value when the time (month) is 0 is 16 inches.
• Hence, the linear function is f(x) = ½x + 16, which corresponds to option A.
• The equation of a line that contains the points (5, 0) and (5, −2) is vertical because both points have the same x-coordinate.
• A vertical line's equation is x = some constant value, which in this case is x = 5, corresponding to option A.
• To find the equation that represents a line passing through points (−1, 2) and (3, 1), we can use the two-point formula or slope-intercept form, but by checking the given options, we see that option A fits the points: 4x − y = −6.
• In the scenario of Jewels riding the ferry, the independent variable is the number of rides she takes, and the dependent variable is the total cost, which depends on the number of rides taken.
• This corresponds to option A.

Answer 2

Part 1) Option A. f of x equals one half times x plus 16

Part 2) Option A. x = 5

Part 3) Option C. x − 4y = −9

Part 4) Option A. The number of rides is the independent variable, and the total cost is the dependent variable.

Explanation:

Part 1)

Let

x -----> the number of months

y ----> vertical jump in inches

step 1

Find the slope

we have the points

(0,16) and (2,17)

`m=(17-16)/(2-0)=(1)/(2)`

The equation of the line in slope intercept form is

`y=mx+b`

we have

`m=(1)/(2)`

`b=16` -----> the point (0,16) is the y-intercept

substitute

`y=(1)/(2)x+16`

convert to function notation

f(x)=y

`f(x)=(1)/(2)x+16`

Part 2) What is the equation of a line that contains the points (5, 0) and (5, −2)?

step 1

Find the slope

we have the points

(5, 0) and (5, −2)

`m=(-2-0)/(5-5)=(-2)/(0)`

the slope is undefined

This is a vertical line (parallel to the y-axis)

therefore

The equation is

x=5

Part 3) Choose the equation that represents a line that passes through points (−1, 2) and (3, 1)

step 1

Find the slope

we have the points

(−1, 2) and (3, 1)

`m=(1-2)/(3+1)=-(1)/(4)`

step 2

Find the equation of the line into point slope form

`y-y1=m(x-x1)`

we have

`m=-(1)/(4)`

`point\ (3, 1)`

substitute

`y-1=-(1)/(4)(x-5)`

Convert to standard form

Multiply by 4 both sides to remove the fraction

`4y-4=-(x-5)`

`4y-4=-x+5`

`x-4y=-4-5`

`x-4y=-9`

Part 4) Jewels has $6.75 to ride the ferry around Connecticut. It will cost her $0.45 every time she rides. Identify the dependent variable and independent variable in this scenario

we know that

The independent variable is the variable whose change isn’t affected by any other variable (An example the age and time)

The dependent variable it’s what changes as a result of the changes to the independent variable (An example of a dependent variable is how tall you are at different ages. The dependent variable (height) depends on the independent variable (age))

Let

x ----> the number of rides

y ----> the total cost in dollars

In this problem

The independent variable or input is the number of rides

The dependent variable or output is the total cost