interpreting a linear function
interpret the linear function to solve the problems. show your work.
1 a group of volunteers is spending a week
cleaning up the trails in the hudson highlands.
on day 2 the volunteers begin at the point on the
trail where they ended the day before. the graph
shows their elevation, in feet, as a function of the
number of hours they work to clean the trails.

a. what does the ordered pair (1, 1000) on the graph represent?

b. the graph begins at 720 on the y - axis. what does this value represent? is this
the rate of change or the initial value?

c. by how many feet does the elevation increase for one hour of work? what does
this value represent, rate of change or initial value?

d. what is the equation that represents this function?

2 the table shows number of people as a
function of time in hours. write an equation for
the function and describe a situation that it
could represent. include the initial value, rate
of change, and what each quantity represents
in the situation.

| hours | number of people |
|-------|------------------|
| 1 | 150 |
| 3 | 250 |
| 5 | 350 |

Question

interpreting a linear function
interpret the linear function to solve the problems. show your work.
1 a group of volunteers is spending a week
cleaning up the trails in the hudson highlands.
on day 2 the volunteers begin at the point on the
trail where they ended the day before. the graph
shows their elevation, in feet, as a function of the
number of hours they work to clean the trails.

a. what does the ordered pair (1, 1000) on the graph represent?

b. the graph begins at 720 on the y - axis. what does this value represent? is this
the rate of change or the initial value?

c. by how many feet does the elevation increase for one hour of work? what does
this value represent, rate of change or initial value?

d. what is the equation that represents this function?

2 the table shows number of people as a
function of time in hours. write an equation for
the function and describe a situation that it
could represent. include the initial value, rate
of change, and what each quantity represents
in the situation.

| hours | number of people |
|-------|------------------|
| 1     | 150              |
| 3     | 250              |
| 5     | 350              |

Accepted Answer

### Part 1: Problem 1 (Interpreting Linear Function for Trail Cleaning)

#### a. Interpreting (1, 1000)
# Brief Explanations:
The x - axis is time in hours, and the y - axis is elevation in feet. The ordered pair (1, 1000) means that when the volunteers have worked for 1 hour, their elevation is 1000 feet.
# Answer:
The ordered pair (1, 1000) represents that after working for 1 hour, the volunteers' elevation is 1000 feet.

#### b. Interpreting the y - intercept (720)
# Brief Explanations:
The y - axis represents elevation. The graph begins at 720 on the y - axis, which is when the time (x - value) is 0. So this is the elevation at the start (when they begin working, 0 hours of work done). The initial value of a function is the value when the independent variable (time here) is 0, while the rate of change is the slope. So 720 is the initial value.
# Answer:
The value 720 represents the elevation of the volunteers when they start working (at 0 hours of work). This is the initial value.

#### c. Finding the rate of change (elevation increase per hour)
# Explanation:
## Step 1: Identify two points
We know the y - intercept is (0, 720) and we have the point (1, 1000) from part a.
## Step 2: Calculate the slope (rate of change)
The formula for slope (rate of change) $m=\frac{y_2 - y_1}{x_2 - x_1}$. Using $(x_1,y_1)=(0,720)$ and $(x_2,y_2)=(1,1000)$, we get $m=\frac{1000 - 720}{1 - 0}=\frac{280}{1}=280$.
This value represents the rate of change because it shows how much the elevation (dependent variable) changes per unit of time (independent variable, 1 hour).
# Answer:
The elevation increases by 280 feet for one hour of work. This value represents the rate of change.

#### d. Equation of the linear function
# Explanation:
## Step 1: Recall the slope - intercept form
The slope - intercept form of a linear function is $y = mx + b$, where $m$ is the slope (rate of change) and $b$ is the y - intercept (initial value).
## Step 2: Substitute $m$ and $b$
We found $m = 280$ (from part c) and $b = 720$ (from part b). So the equation is $y=280x + 720$, where $y$ is the elevation in feet and $x$ is the time in hours.
# Answer:
The equation of the function is $y = 280x+720$, where $x$ is the number of hours worked and $y$ is the elevation in feet.

### Part 2: Problem 2 (Table of People as a Function of Time)

#### Step 1: Find the rate of change (slope)
# Explanation:
## Step 1: Select two points from the table
Let's take the points (1, 150) and (3, 250). The formula for slope $m=\frac{y_2 - y_1}{x_2 - x_1}$.
## Step 2: Calculate the slope
$m=\frac{250 - 150}{3 - 1}=\frac{100}{2}=50$.
#### Step 2: Find the initial value (b)
Use the slope - intercept form $y=mx + b$ and the point (1, 150). Substitute $x = 1$, $y = 150$ and $m = 50$ into the equation: $150=50	imes1 + b$.
## Step 3: Solve for b
$150=50 + b$, so $b=150 - 50 = 100$.
#### Step 3: Write the equation
The equation is $y = 50x+100$, where $x$ is the number of hours and $y$ is the number of people.
#### Step 4: Describe the situation
The initial value $b = 100$ could represent the number of people at time 0 (e.g., the number of people present before any time has passed, like at the start of an event). The rate of change $m = 50$ means that 50 more people arrive each hour. For example, this could represent the number of people at a concert, where 100 people are already there when the event starts, and 50 people arrive each hour.
# Answer:
The equation of the function is $y = 50x + 100$.
Situation: Consider a gathering (like a festival) where initially (at 0 hours) there are 100 people. Then, 50 people join every hour. Here, $x$ is the number of hours since the start of the gathering, $y$ is the total number of people, the initial value (100) is the number of people at the start, and the rate of change (50) is the number of people joining per hour.

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QUESTION IMAGE

Question

Explanation:

Part 1: Problem 1 (Interpreting Linear Function for Trail Cleaning)

a. Interpreting (1, 1000)

Step 1: Identify two points

Step 2: Calculate the slope (rate of change)

Answer:

b. Interpreting the y - intercept (720)