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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

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:
\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 expressions?

### Interpreting exponential expression word problem

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 ?
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.)

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?

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?

## Want to join the conversation?

• Can u put some more examples regarding exponential expressions?
• 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?
• 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.