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Quadratic and exponential word problems | Lesson

A guide to quadratic and exponential word problems on the digital SAT

What are quadratic and exponential word problems?

Both quadratic functions and exponential functions can be used to model nonlinear relationships in everyday life, such as the height of a falling object or the population change of a city.
In this lesson, we'll learn to:
  1. Interpret quadratic and exponential functions
  2. Solve quadratic and exponential word problems
  3. Write quadratic and exponential functions, including rewriting exponential functions using different time units
Note: An understanding of percentages is useful for interpreting exponential functions.
You can learn anything. Let's do this!

What are some common SAT scenarios modeled by quadratic functions?

Interpret quadratic models: Factored form

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Interpret quadratic models: Factored formSee video transcript

Area of a rectangle

The formula for A, the area of a rectangle with length ell and width w is:
A, equals, ell, w
In a quadratic function dealing with area, the area is the output, one of the linear dimensions is the input, and the other linear dimension is described in terms of the input. The quadratic expression is usually written in factored form, with the length and width represented by a factors.
For example, if the length of a rectangular piece of paper is x inches and the width is 2 inches shorter than the length, then the area of the rectangle is equal to:
A=w=x(x2)\begin{aligned} A &= \ell w \\\\ &=x(x-2) \end{aligned}

Height versus time

The quadratic function h for the height of an object at time t looks like a quadratic function in standard form:
h, left parenthesis, t, right parenthesis, equals, minus, a, t, squared, plus, b, t, plus, c
Where:
  • t is the input variable. It usually represents time in seconds.
  • c describes the initial height of the object, or the object's height when t, equals, 0.
For example, if the height of a projectile in feet is modeled by the function h, left parenthesis, t, right parenthesis, equals, minus, 16, t, squared, plus, 144, t, plus, 32, where t is time in seconds:
  • The initial height of the projectile is 32 feet because the constant term is equal to 32.

Try it!

Try: find the length of a rectangle
A, equals, left parenthesis, x, right parenthesis, left parenthesis, 3, x, minus, 2, right parenthesis
The equation above models A, the area of a rectangle with a width of x meters.
represents the length of the rectangle in meters.
According to the model, the length of the rectangle is
.


What are some common SAT scenarios modeled by exponential functions?

Interpreting exponential expression word problem

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Interpreting exponential expression word problemSee video transcript

Population growth and decline

The exponential function P for population looks like the following:
P, left parenthesis, t, right parenthesis, equals, P, start subscript, 0, end subscript, r, start superscript, t, end superscript
Where:
  • t is the input variable representing the number of time periods elapsed.
  • P, start subscript, 0, end subscript is the initial population, or the population when t, equals, 0.
  • r describes how the population is changing.
For example, if P, left parenthesis, t, right parenthesis, equals, 75, left parenthesis, 1, point, 04, right parenthesis, start superscript, t, end superscript describes the population of a village t years after 2010:
  • t represents the number of years after 2010. In the year 2011, t, equals, 1; in the year 2020, t, equals, 10.
  • The initial population of the village is 75.
  • 1, point, 04 tells us that for each year after 2010, the population of the town is 1, point, 04 times the population in the previous year.
If we convert 1, point, 04 to 104, percent, we can also say that the population of the town grows by 104, percent, minus, 100, percent, equals, 4, percent each year.
If r, is greater than, 1, then the population is growing. If 0, is less than, r, is less than, 1, then the population is declining.

Compounding interest

The exponential function P for an amount of money accruing compounding interest looks like the following:
P, left parenthesis, t, right parenthesis, equals, P, start subscript, 0, end subscript, left parenthesis, 1, plus, r, right parenthesis, start superscript, t, end superscript
Where:
  • t is the input variable representing the number of time periods elapsed.
  • P, start subscript, 0, end subscript is the initial amount of money, or the amount of money before any interest is accrued.
  • r is the interest rate applied for each time period expressed as a decimal.
Note: there is a more complex version of the formula in which the interest can be applied multiple times within a single time period (for example, an annual interest rate with monthly interest calculations), but that version typically does not appear on the SAT.
For example, if P, left parenthesis, t, right parenthesis, equals, 500, left parenthesis, 1, point, 01, right parenthesis, start superscript, t, end superscript models the amount of money, in dollars, in a savings account after t years:
  • t represents the number of years after the initial deposit.
  • 500 represents the initial amount put into the savings account: 500 dollars.
  • 1, point, 01, equals, 1, plus, 0, point, 01, which means r, equals, 0, point, 01 and the (annual) interest rate is 1, percent.

Try it!

Try: interpret population change
P, left parenthesis, t, right parenthesis, equals, 680, left parenthesis, 0, point, 94, right parenthesis, start superscript, t, end superscript
The function above models the population of a rural village t years after 1996.
Based on the function, what was the village's population in 1996 ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
By what percent is the village's population declining each year? (Ignore the percent symbol when entering your answer. For example, if the answer is 12, percent, enter 12.)
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


How do I solve quadratic and exponential word problems?

Quadratic word problem: ball

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Quadratic word problem: ballSee video transcript

Exponential expressions word problems (algebraic)

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Exponential expressions word problems (algebraic)See video transcript

Common word problem scenarios on the SAT

Area of a rectangle

Because the formula for the area of a rectangle, A, equals, ell, w, is commonly known, you're expected to be able to write quadratic equations modeling rectangular areas and solve for length or width when area is given.

Height versus time

You're not expected to know the physics of falling objects; functions that describe the relationship between height and time will be given to you. However, you might be asked to:
  • Calculate the height of the object at a given time
  • Calculate the time at which the object is at a given height
A common given height is "the ground", which means a height of 0 units.

Example:
h, left parenthesis, t, right parenthesis, equals, minus, 16, t, squared, plus, 64, t
The function above models the height h, in feet, of an object above ground t seconds after being launched straight up in the air. At how many seconds after launch does the object fall back to the ground?

Population growth and decay / Compounding interest

For both of these topics, you'll either be asked to write a function based on a verbal description or to evaluate the function when one is given. The SAT generally doesn't ask you to do both in the same question.
Functions modeling these topics typically look like f, left parenthesis, t, right parenthesis, equals, a, left parenthesis, b, right parenthesis, start superscript, t, end superscript, where a is the initial value, b describes
change
, and t is the variable representing time.

Example:
P, left parenthesis, t, right parenthesis, equals, 227, left parenthesis, 1, point, 03, right parenthesis, start superscript, t, end superscript
The function above models P, the population of Pallet Town, t years after 1996. To the nearest whole number, what was the net increase of Pallet Town's population from 1996 to 2002 ?

Try it!

try: write a quadratic equation
The length of a rectangular plot of land is 3 times its width. The area of the plot is 1, point, 08 square miles.
If we use x to represent the width of the plot in miles, then the length of the plot is
miles.
Write an equation that gives us the width of the plot when solved.


try: write an exponential expression
Yuji put dollar sign, 500 into a savings account that yields 1, percent interest annually.
The initial amount of money in the savings account is dollar sign
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.
After one year, the amount of money in the account is dollar sign
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
, or 1, point, 01 times the initial amount.
After two years, the amount of money in the account is dollar sign
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
, or 1, point, 01 times 1, point, 01 times the initial amount.
Write an expression that describes the amount of money in the account after t years.


How do I rewrite exponential functions using different time units?

Interpreting time in exponential models

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Interpreting time in exponential modelsSee video transcript

How do we change time units in an exponential expression?

On the SAT, we're sometimes given an exponential model in one time unit (for example: years), and asked to re-write it using another time unit (for example: months), showing the same rate of growth or decay as the original model.
For example, let's look at the equation M, equals, 100, left parenthesis, 1, point, 62, right parenthesis, start superscript, t, end superscript, where M is the number of members a club has t years after it opens. How do we write an equation for the number of club members m months after opening?
In this type of question, the base of the exponent is kept the same, and only the exponent changes. We know that the number of club members increases by a factor of 1, point, 62 after t, equals, 1 year, which means the number of club members also needs to increase by a factor of 1, point, 62 after m, equals, 12 months.
To make sure the exponent of 1, point, 62 is 1 when m, equals, 12 and 2 when m, equals, 24 (two years), we need to divide m by 12 in the exponent:
M, equals, 100, left parenthesis, 1, point, 62, right parenthesis, start superscript, start superscript, start fraction, m, divided by, 12, end fraction, end superscript, end superscript
When changing the time units in an exponential expression:
  • Identify the unit equivalence used for the question, e.g., 1, start text, y, e, a, r, end text, equals, 12, start text, m, o, n, t, h, s, end text.
  • Tip: If converting the variable from a larger time unit to a smaller time unit, we need to make the exponent smaller. For example, when converting from years to months, we divide the variable representing the number of months by 12.
  • Tip: If converting the variable from a smaller time unit to a larger time unit, we need to make the exponent larger. For example, when converting from months to years, we multiply the variable representing the number of years by 12.
  • Verify that the two expressions give us the same values when evaluated using equivalent amounts of time, e.g., 1 year and 12 months, 2 years and 24 months, etc.

Try it!

Try: convert from hours to days
P, equals, 50, left parenthesis, 1, point, 1, right parenthesis, start superscript, h, end superscript
The equation above models the population, P, of a bacteria culture after h hours of incubation.
There are 24 hours in a day. Since a day is
than an hour, if we want to write an equation that models the population of the bacteria culture after d days of incubation, we must
the exponent by 24.
Which of the following equations models the population of the bacteria culture after after d days of incubation?
Choose 1 answer:
Choose 1 answer:


Your turn!

Practice: interpret the meaning of a constant
h, left parenthesis, t, right parenthesis, equals, minus, 4, point, 9, t, squared, plus, 7, point, 7, t, plus, 0, point, 5
The function above models the height h, in meters, of a soccer ball above ground t seconds after being kicked by a soccer player. What does the number 0, point, 5 represents in the function?
Choose 1 answer:
Choose 1 answer:


Practice: determine the value of a constant
The value of a rare baseball card was dollar sign, 80 the year it was released. Pepe estimates that the value of the card increases by 7, percent each year and uses the expression 80, left parenthesis, x, right parenthesis, start superscript, t, end superscript to estimate the value of the card t years after its release. What is the value of x in the expression?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


practice: calculate compounding interest
Janice uses the expression 2300, left parenthesis, 1, point, 058, right parenthesis, start superscript, t, end superscript to model the amount of money in her investment account after t years. To the nearest whole dollar, how much money is in Janice's investment account after 10 years? (Ignore the dollar sign when entering your answer)
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Practice: write population function
A gym currently has 5, comma, 500 members. Due to an economic downturn, an analyst predicts that the gym will lose 2, point, 5, percent of its members each month for the foreseeable future. Which of the following functions models P, left parenthesis, t, right parenthesis, the number of gym members t months from now?
Choose 1 answer:
Choose 1 answer:


Practice: solve a word problem representing an uncommon scenario
p, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 0, point, 6, right parenthesis, left parenthesis, 300, minus, 75, x, right parenthesis
Roy uses the function above to model p, the daily profit in dollars, of his taco truck from selling tacos at x dollars each. Based on the model, at which of the following prices would Roy be able to earn the most profit?
Choose 1 answer:
Choose 1 answer:


Practice: change the time unit in an exponential model
D, equals, 200, left parenthesis, 1, point, 16, right parenthesis, start superscript, m, end superscript
The equation above models the number of total downloads, D, for an app Clara created m months after its launch. Of the following, which equation models the number of total downloads y years after launch?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • starky ultimate style avatar for user scorp10
    Sal does Interpreting-time-in-exponential-models question wrong. He solves for 9/5, while he must do so for 4/5.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Madhih
      he solved for 9/5 because the question is when the tree gains 4/5 of the initial amount of branches. lets take the initial amount of branches as 1 or 5/5. so, when the tree gains 4/5 of the initial amount of branches we have to add the initial amount and the amount of branches gained to get the current amount that is 9/5. so, when the tree gains 4/5 branches the total amount of branches would be 9/5. so, when solving for 9/5 we can find when the tree gained 4/5 branches.
      (2 votes)