Main content

## SAT

### Unit 6: Lesson 4

Passport to Advanced Math: lessons by skill- Solving quadratic equations | Lesson
- Interpreting nonlinear expressions | Lesson
- Quadratic and exponential word problems | Lesson
- Manipulating quadratic and exponential expressions | Lesson
- Radicals and rational exponents | Lesson
- Radical and rational equations | Lesson
- Operations with rational expressions | Lesson
- Operations with polynomials | Lesson
- Polynomial factors and graphs | Lesson
- Graphing quadratic functions | Lesson
- Graphing exponential functions | Lesson
- Linear and quadratic systems | Lesson
- Structure in expressions | Lesson
- Isolating quantities | Lesson
- Function Notation | Lesson

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Quadratic and exponential word problems | Lesson

## What are quadratic and exponential word problems, and how frequently do they appear on the test?

**Quadratic and exponential word problems**ask us to solve equations or evaluate functions that model real-world scenarios using quadratic and exponential expressions.

On your official SAT, you'll likely see

**1 to 2 questions**that are quadratic or exponential word problems.This lesson builds on an understanding of the following skills:

**You can learn anything. Let's do this!**

## How do I solve quadratic and exponential word problems?

### Quadratic word problem: ball

### Exponential expressions word problems (algebraic)

### Common word problem scenarios on the SAT

The

**Interpreting nonlinear expressions**lesson details how to interpret quadratic and exponential expressions modeling common scenarios that appear on the SAT. The four common word problem scenarios are:- Area of a rectangle (quadratic)
- Height versus time (quadratic)
- Population growth and decay (exponential)
- Compounding interest (exponential)

#### 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:**

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 , and t is the variable representing time.**Example:**

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!

## Your turn!

## Want to join the conversation?

- Uhm this is probs a dumb question but why on the population growth/decay example the p(t) becomes p(4) instead of p(6) (the amount of years since 1996)?(14 votes)
- Looks like just a simple typo to me, good catch! If you want, you can tell the folks at KA to fix it by clicking the "Report a mistake" link right below where you go to ask a question. This way, other learners won't be confused either.(15 votes)

- In the ball qns (video), how does the 50 Ft above ground affect the equation? Why has Sal not taken that into account?(2 votes)
- The 50 feet that the ball has for an initial height does affect the equation, you're right. However, based on the way the question is worded, we can trust that the given equation will represent that specific ball's height as a function of time, when it started up 50 feet and was shot with an initial velocity of 20 feet/sec.

The "+50" term in the equation basically accounts for the added height. If you're curious about how the equation is derived in general, the Khan Academy videos on projectile motion in the physics playlist offer a great starting point.(3 votes)

- In the first video at1:25, why can you divide it all by -2? I never would have thought to do that.(3 votes)
- You can divide it all by -2 because 2 is a common factor between all numbers in the equation. 0, -16, 20, and 50 can all be divided by 2 to equal a whole number. The reason the 2 was switched to -2 was to make -16 positive, since it was the leading number in the equation.(1 vote)

- For the last example (Roy), what would be the way to actually solve it instead of just plugging in the answers?(1 vote)
- The fastest way to solve it is to average the x-intercepts. This gives you the x coordinate for the vertex, which is 2.3.

Of the 4 answer choices given, choice B is closest to 2.3. That's the correct answer. 🙂(2 votes)

- I think there is a calculation mistake in the question where janice is calculating her investment after 10 years. The question says that2300(1.058)10 is equal to 4041.89, which is incorrect calculation wise. It is equal to 4048 approx. Please let me know if I am incorrect.(1 vote)
- The calculation is correct, you just made a mistake while rounding (if your number is followed by 5, 6, 7, 8, or 9 after the decimal dot, round the number up for example 23.9 would be 24) If your number is not followed by any of those you don't round it up :)) hope it helps someone.(2 votes)