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## SAT

### Course: 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

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# Interpreting nonlinear expressions | Lesson

## What does "interpreting nonlinear expressions" mean, and how frequently do these questions appear on the test?

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. The ability to interpret these functions will allow us to better understand these scenarios.**Note:**An understanding of percentages is useful for interpreting exponential expressions.

On your official SAT, you'll likely see

**1 question**that tests your ability to interpret nonlinear expressions. In addition, understanding the scenarios explored in this lesson will help you solve quadratic and exponential word problems.**You can learn anything. Let's do this!**

## What are some common SAT scenarios modeled by quadratic expressions?

### Interpret quadratic models: Factored form

### Area of a rectangle

The formula for A, the area of a rectangle with length ell and width w is:

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:

### Height versus time

The quadratic function h for the height of an object at time t looks like a quadratic function in

**standard form**: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!

## What are some common SAT scenarios modeled by exponential expressions?

### Interpreting exponential expression word problem

### Population growth and decline

The exponential function P for population looks like the following:

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:

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!

## Your turn!

## Want to join the conversation?

- Can u put some more examples regarding exponential expressions?(36 votes)
- In the equation "A=(x)(3x-2)" it asks what represents the length. Based on the equation for area being length times width. I reasoned that "x" represents the length due to the order of the equation. Though it says 3x-2 represents the length. Am I wrong?(4 votes)
- Multiplication is commutative, meaning that yo can switch around the positions of factors all you like and the answer will stay the same. For this reason, we cant assume that the equation will be in the same order as the question describes it. A = l * w = w * l. Instead, the question tells us that the area describes a rectangle with a width of x meters, so x has to be the width. From our formula, the other factor, 3x - 2, is then the length.(5 votes)

- when i was practicing nonlinear equations on KA i faced some questions about the horizontal/vertical asymptote of the equation y= a/x-k +h. do these questions come on the SAT? and where can i study them?(2 votes)
- Questions that ask you about the asymptotes of a rational equation will be really quite rare, but I think that it is in the scope of the math test, fitting in under the Passport to Advanced Math portion of it. To study, you'll probably really only need to know really basic things, such as how to find the asymptote. Khan Academy's playlist of Rational functions should really have everything. Good luck.

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:rational#x2ec2f6f830c9fb89:rational-graphs(5 votes)

- about that last

practice question...is it that the 7% of the initial value or the value of the previous year.(1 vote)- If a value increases by 7% each year, it's value for the current year will be the previous year's value plus a 7% increase, or the previous year's value multiplied by 1.07. If the value of the current year was 7% of the value of the current year, that wouldn't make sense as you're defining your relationship with the value you're defining.(2 votes)

- I still don't get why we had to add 1 to r to find x. Please can you explain it deeper.(1 vote)
- Don’t have any(0 votes)