Question

4.1 Writing Equations in Slope-Intercept Form (continued) 2 EXPLORATION: Mathematical Modeling Work with a partner. The graph shows the cost of a smartphone plan. Smartphone Plan a. What is the y-intercept of the line? Interpret the y-intercept in the context of the problem Cost per month (dollars) y 100 80 60 40 20 0 0 500 1000 1500 2000 250 Data usage (megabyte b. Approximate the slope of the line.

Answer 1

a) y-intercept = 20 dollars

In the context of the question, this means that the cost per month of the data usage on that smartphone is 20 dollars.

b) Slope = 0.03 dollars per megabyte

In the context of the question, this means that the cost of data per megabyte is $0.03

c) y = 0.03x + 20

Check Explanation for the second question.

Step-by-step explanation

a) The y-intercept is the point where the line crosses the y-axis, that is, the value of y when x = 0

From the attached graph, we can see that the graph crosses the y-axis at

y = 20 dollars

In the context of the question, this means that the cost per month of the data usage on that smartphone is 20 dollars.

b) For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are (x₁, y₁) and (x₂, y₂), the slope is given as

`Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=(y_2-y_1)/(x_2-x_1)`

For this question, we will pick two points

(x₁, y₁) and (x₂, y₂) are (0, 20) and (2000, 80)

`\text{Slope = }(80-20)/(2000-0)=(60)/(2000)=\text{ 0.03}`

Slope = 0.03 dollars per megabyte

In the context of the question, this means that the cost of data per megabyte is $0.03

c) Since this is a straight line,

The slope and y-intercept form of the equation of a straight line is given as

y = mx + c

where

y = y-coordinate of a point on the line.

m = slope of the line.

x = x-coordinate of the point on the line whose y-coordinate is y.

c = y-intercept of the line.

For this question,

y = Cost of data per month

m = slope = 0.03 dollars per megabyte

x = data usage in megabytes

c = y-intercept = 20

y = 0.03x + 20

For the second question,

We are asked to find the slope and y-intercept of each line

We've explained how to get the slope and y-intercept the other time

a) (x₁, y₁) and (x₂, y₂) are (0, -1) and (2, 3)

`\text{Slope = }(3-(-1))/(2-0)=(3+1)/(2)=(4)/(2)=2`

And we can evidently see that the line crosses the y-axis at the point y = -1. So, y-intercept = c = -1

The slope and y-intercept form of the equation of a straight line is given as

y = mx + c

where

y = y-coordinate of a point on the line.

m = slope of the line = 2

x = x-coordinate of the point on the line whose y-coordinate is y.

c = y-intercept of the line = -1

y = 2x - 1

b) (x₁, y₁) and (x₂, y₂) are (4, -2) and (0, 2)

`\text{Slope = }(2-(-2))/(0-4)=(2+2)/(-4)=(4)/(-4)=-1`

And we can evidently see that the line crosses the y-axis at the point y = 2. So, y-intercept = c = 2

The slope and y-intercept form of the equation of a straight line is given as

y = mx + c

where

y = y-coordinate of a point on the line.

m = slope of the line = -1

x = x-coordinate of the point on the line whose y-coordinate is y.

c = y-intercept of the line = 2

y = -x + 2

c) (x₁, y₁) and (x₂, y₂) are (-3, 3) and (3, -1)

`\text{Slope = }(-1-3)/(3-(-3))=(-4)/(3+3)=(-4)/(6)=-(2)/(3)`

To find the y-intercept for this, we will use one of the given points and (0, c) to find c

(x₁, y₁) and (x₂, y₂) are (-3, 3) and (0, c)

Slope = -(2/3)

`\begin{gathered} \text{Slope = }(c-3)/(0-(-3)) \\ -(2)/(3)=(c-3)/(0+3) \\ -(2)/(3)=(c-3)/(3) \\ \text{Cross multiply} \\ 3(c-3)=3(-2) \\ 3c-9=-6 \\ 3c=-6+9 \\ 3c=3 \\ \text{Divide both sides by 3} \\ (3c)/(3)=(3)/(3) \\ c=1 \end{gathered}`

The slope and y-intercept form of the equation of a straight line is given as

y = mx + c

where

y = y-coordinate of a point on the line.

m = slope of the line = -(2/3)

x = x-coordinate of the point on the line whose y-coordinate is y.

c = y-intercept of the line = 1

y = (-2/3)x + 1

d) (x₁, y₁) and (x₂, y₂) are (4, 0) and (2, -1)

`\text{Slope = }(-1-0)/(2-4)=(-1)/(-2)=(1)/(2)`

To find the y-intercept for this, we will use one of the given points and (0, c) to find c

(x₁, y₁) and (x₂, y₂) are (4, 0) and (0, c)

Slope = (1/2)

`\begin{gathered} \text{Slope = }(c-0)/(0-4) \\ (1)/(2)=(c)/(-4) \\ \text{Cross multiply} \\ 2c=-4 \\ \text{Divide both sides by 2} \\ (2c)/(2)=-(4)/(2) \\ c=-2 \end{gathered}`

The slope and y-intercept form of the equation of a straight line is given as

y = mx + c

where

y = y-coordinate of a point on the line.

m = slope of the line = (1/2)

x = x-coordinate of the point on the line whose y-coordinate is y.

c = y-intercept of the line = -2

y = (½)x - 2

Hope this Helps!!!