Unit 5 Functions — Unit Plan

A parent is using a garden hose to fill up a small inflatable pool for her young child. The pool has a capacity of 90 gallons. She turns the water off after 5 minutes but leaves the hose in the pool for another 3 minutes before putting it away.

In this situation, the relationship between the gallons of water in the pool and the time since the parent started filling the pool can be seen as a function.

1. In that function, which variable is independent? Which one is dependent?
2. Write a sentence of the form “$\underline{\hspace{0.5in}}$ is a function of $\underline{\hspace{0.5in}}$.”

3. Sketch a possible graph of the relationship on the coordinate plane. Be sure to label and indicate a scale on each axis.

   

Function $Q$ gives a puppy’s weight in pounds as a function of its age in months.

1. What does each expression or equation represent in this situation?

   1. $Q(18)$
   2. $Q(30)= 27.5$

2. Use function notation to represent each statement.

   1. When the puppy turned 12 months old, it weighed 19.6 pounds.
   2. When the puppy was $m$ months old, it weighed $w$ pounds.

An art museum opens at 9 a.m. and closes at 5 p.m. The function $V$ gives the number of visitors in a museum $h$ hours after it opens.

1. Explain what this statement tells us about the situation: $V(1.25) = 28$.

2. Use function notation to represent each statement:

   1. At 1 p.m., there were 257 visitors in the museum.
   2. At the time of closing, there were no visitors in the museum.

3. Use the previous statements about the visitors in the museum to sketch a graph that could represent the function.

   

1. Complete the table with the perimeter of a square for each given side length.

   side length (inches)	perimeter (inches)
   0.5
   7
   20

2. Write a rule for a function, $P$, that gives the perimeter of a square in inches when the side length is $x$ inches.
3. What is the value of $P(9.1)$? What does it tell us about the side length and perimeter of the square?

Elena, who is looking for a gaming console, found a third option, which also offers a 1 month free trial period of the online service.

The monthly cost of this option can be represented by function $C$, defined by $C(x) = 5x + 260$, where $x$ is the number of months Elena uses the online gaming service after the free trial period.

1. Find $C(1)$, and explain what it means in this situation.
2. Elena’s budget is still $280 a month. To find out how many months of the service Elena gets, she writes $C(x) = 280$ and solves for $x.$ What value of $x$ makes this equation true? Show your reasoning.

A squirrel runs up and down a tree.

The graph shows the height of the squirrel, $h$ (feet), as a function of time, $t$ (seconds).

1. What is the highest point the squirrel reaches?
2. Solve $h(t)=0$. What does this solution tell you about the squirrel?
3. Find the vertical intercept of the graph. What does it tell you about the squirrel?

1. The graph shows the population of a city from 1900 to 2000.

   What is the average rate of change of the population between 1930 and 1950? Show your reasoning.

   

   A graph, origin O. Horizontal axis, year, scale 1900 to 2000 by 10. Vertical axis, population in thousands, scale 0 to 50 by 5s. The curve passes through the points 1900 comma 20, 1910 comma 22, 1920 comma 32, 1930 comma 40, 1940 comma 44, 1950 comma 45, 1960 comma 43, 1970 comma 35, 1980 comma 30, 1990 comma 28, and 2000 comma 26.

2. For each interval, decide if the average rate of change is positive or negative.

   1. from 1930 to 1940
   2. from 1950 to 1970
   3. from 1930 to 1970
3. In which decade (10-year interval) did the population grow the fastest? Explain how you know.

Functions $R$ and $D$ give the height, in feet, of a toy rocket and a drone, $t$ seconds after they are released. Here are the graphs of $R$ (for the rocket) and $D$ (for the drone).

Two functions graphed on a coordinate plane, no grid, origin O. Horizontal axis, time in seconds, from 0 to 7 by 1’s. Vertical axis, height in feet, from 0 to 50, by 10’s. Function R, curved, starts at 0 comma 25, increases until 2 comma 45, decreases until 5 comma 0. Function D, piecewise function, starts at origin, increases until 2 comma 20, horizontal line until 5 point 25 comma 20, decreasing until 7 comma 0.

1. Which of the inequalities is true: $R(2)>D(2)$ or R(2)<d></d></span>? </li> <li>What was the height of the drone when the toy rocket hit the ground? </li> <li>For what value of <span class="math">\(t is $R(t)=D(t)$ true? What does this tell you about the drone and the toy rocket?

Diego's club earns money for charity when members of the club perform community service after school. For each student who does community service, the club earns $5. There are 12 students in the club.

The total dollar amount earned, $E$, is a function of the number of members who perform community service, $n$.

1. Is 5 a possible input value? Why or why not?
2. Is 24 a possible output value? Why or why not?
3. Describe the domain of this function.
4. Describe the range of this function.

If you get stuck, consider creating a table or a graph.

The function $W$ gives the temperature, in degrees Fahrenheit, of a pot of water on a stove $t$ minutes after the stove is turned on. The water boils at $212^\circ \text{F}$, and the pot is taken off the stove.

The graph of the function is shown.

1. Is 250 in the range of function $W$? Explain how you know.
2. Describe the range of the function.
3. Does $W(t)=0$ have a solution? Explain how you know.

$P$ is a function that gives the cost, in dollars, of mailing a letter from the United States to Mexico in 2023 based on the weight of the letter in ounces, $w$.

1. How much does it cost to send a letter that weighs 1.5 ounces? 2 ounces?
2. Sketch a graph of the function on the coordinate plane.

Bags of almonds from a food producer are advertised to weigh 500 grams each. Twenty bags of almonds are weighed.

The scatter plot shows the absolute error between the actual weight and the advertised weight of those bags.

1. Estimate the largest absolute error. What is the weight of that bag?
2. Another bag weighs 495.3 grams. What is its absolute error?

horizontal axis, weight in grams. vertical axis, absolute error in grams. scatterplot with vertex at 500 comma 0. most points on the scatter plot have an absolute error greater than 4 grams.

The term "elevation" is often used to describe the height of a place (such as a city, a mountain, or a valley) compared to sea level. For example, the highest point of Houston, Texas has an elevation of 105 feet. The surface of the sea has an elevation of 0 feet. Some places are below sea level, so their elevations are negative values.

1. The table shows the elevation, $e$, of several towns.

   Function $f$ gives the vertical distance of each town from sea level. Both $e$ and $f(e)$ are measured in feet. Complete the table of values.

   $e$	180	12.1	5.4		-5.4	-36	-180
   $f(e)$				0

2. Write an equation to represent $f(e)$.

3. Two towns have different elevations, but when the elevations are used as inputs of $f(e)$, they both produce an output of 25.

   What are the elevations of the two towns? Why do they produce the same output?

If we know the temperature in degrees Celsius, $C$, we can find the temperature in kelvin, $K$, using this equation:

$K = C + 273.15$

Admission to a carnival costs $5. Tickets for games at the carnival cost $2 each.

1. The equation $C =5 + 2t$ defines a function in this situation. What does the function represent?
2. What does the inverse of this function represent?
3. Does the equation $t=\dfrac {C-2}{5}$ define the inverse of the function in the first question? Explain or show how you know.

Kiran is about to start hiking on a trail that is 1.8 miles long. In his past hikes on the same trail, he had walked at a constant rate of 2.4 miles an hour, which is 0.04 mile a minute. 

1. What is Kiran's remaining distance on the trail after 10 minutes of hiking?
2. After how many minutes of walking will Kiran have 0.2 mile left on the trail?

3. Function $f$ represents Kiran's remaining distance on the trail, in miles, $t$ minutes after he starts walking.

   Write an equation to represent this function. Use function notation.

4. Write an equation that is the inverse of function $f$. Explain what information it gives us about Kiran’s hike.