Math Study Guide for the SAT Exam

Additional Types of Functions

There are several different types of functions you’ll need to be familiar with and be able to manipulate. Here are the details of the types of functions you may see on the SAT.

Nonlinear Functions

We’ve seen that nonlinear equations produce something other than a straight line on a graph. Nonlinear functions do the same. Here is more information about working with this type of function.

Operations

Even as the expressions get more complicated, the order of operations remains your guiding light. To illustrate these steps, let’s go through an example that involves a binomial multiplied by a trinomial:

\[(x^2 + 3)(x^2 - x -1)\]

Though it might not look like it at first, this is actually a single polynomial. We have to do a little work to get it into the form you learned above. We need to be sure to do operations in the right order, or we will end up with the wrong polynomial. According to PEMDAS, the parentheses come first. To deal with them, we distribute \((x^2 - x - 1)\) through \((x^2 + 3)\). This gives us:

\[(x^2 + 3)(x^2 -x-1) = x^2 (x^2 - x - 1) + 3(x^2 - x - 1)\]

Note: We could also have distributed in the other order and found the same result.

Now, we distribute again, using the rules of exponents to simplify:

\[x^2(x^2 - x - 1) + 3(x^2 - x - 1) = x^4 - x^3 - x^2 + 3x^2 - 3x - 3\]

That takes care of all the parentheses. The next step in PEMDAS is the exponents. However, since there are no exponents that we can evaluate, we move on to the next step, multiplication and division. Again, though, this is not needed. So, we move on to the final step, adding and subtracting like terms to simplify. In this case, the only like terms are the two \(x^2\) terms, so we add them:

\[x^4 - x^3 - x^2 + 3x^2 - 3x - 3 =x^4 - x^3 + 2x^2 - 3x - 3\]

This is the expanded form of the polynomial, just like we saw in the introduction.

Analysis and Rewriting

Polynomials can also be written as a product of their factors. The factors of polynomials are simply smaller polynomials. For example, let’s look at this:

\[x^3 - x^2 + 2x - 2\]

This can be factored. To do so, we’ll first pair any terms that have common factors:

\[x^3 - x^2 + 2x - 2 = (x^3 + 2x) - (x^2 + 2) = x(x^2 + 2) - (x^2 + 2) = (x^2 + 2)(x - 1)\]

It’s not always easy to see what a polynomial’s factors are, especially since we often will have like terms that combine when the polynomial is expanded.

If you’re given one form of a polynomial and need to find the other, you can always use unknown numbers in the unknown form, compare it to the known form, and then deduce what the numbers must have been. For example, suppose we are given this equation:

\[x^3 + ax^2 + 2x - 2=(x^2 + b)(x - 1)\]

where \(a\) and \(b\) are constant numbers that we are asked to find. We can expand the right side:

\[(x^2 + b)(x - 1) = x^3 -x^2 + bx - b\]

This means:

\[x^3 +ax^2 + 2x - 2 = x^3 -x^2 + bx - b\]

Now we can compare coefficients, which involves matching the numbers that are multiplied by the like powers of \(x\). One side has \(a\) multiplied by \(x^2\) and the other has \(-1\), so \(a = -1\). Comparing either of the last two coefficients similarly shows that \(b = 2\).

Dividing Polynomials by Linear Expressions

Linear expressions are just polynomials that don’t go any higher than the first power of \(x\). Some questions in the Passport to Advanced Math section of the SAT may ask you to expand a polynomial divided by a linear expression using polynomial long division, which is similar in spirit to numerical long division. You can use the similarity to help yourself remember the procedure for polynomial division. For example, let’s expand this:

\[\frac{4x^2 + 4x + 3}{2x-1}\]

Begin by considering what must be multiplied by \(2x - 1\) (the divisor) to match \(4x^2\), the first term in the numerator (the dividend):

\[2x \cdot 2x = 4x^2\]

So, the first term in the quotient will be \(2x\). Multiply the divisor by this \(2x\) and subtract this value from the numerator:

\[(4x^2 + 4x + 3) - (4x^2 - 2x)=6x + 3\]

Now, the divisor must be multiplied by \(3\) to divide into \(6x\). Multiplying the denominator by \(3\) gives \(6x - 3\), and because we are subtracting this, the signs switch to give \(-6x + 3\), which, when combined, gives a remainder of \(6\), which is then placed in a fraction as the numerator with the original divisor as the denominator to produce the final quotient and remainder:

\[4x^2 + 4x + 3 \div 2x - 1 = 2x + 3 + \frac{6}{2x-1}\]

Exponential Functions

Anytime a quantity’s rate of change is proportional to the quantity itself, an exponential function describes the situation. This is known as exponential growth, and it is found by multiplying the initial amount (\(a_i\)) by the proportion of change (\(r\)), where time (\(t\)) is the exponent. The result is the current amount (\(a\)). As a formula, this is:

\[a=a_1 \times r^{t}\]

For example, if you earn \(5\%\) interest on an investment every year, and you start by investing \(\$200\), the formula for how much the investment will have grown to in \(t\) years is:

\[200 \times (1 + 0.05)^t\]

Note that the proportion of change is \(1.05\), not \(0.05\). The \(1\) accounts for the amount already in the investment, and the \(0.05\) accounts for the interest.

We can replace time with a different variable in the formula above. As another example, suppose you earn twice as many points in a bean bag toss game for every \(10\) feet farther back you are when throwing, starting with \(10\) points at the closest allowed distance. Then the number of points you could earn with a throw \(x\) feet behind the closest line is:

\[10 \times 2^\frac{x}{10}\]

So, at \(10\) feet you earn \(20\) points, at \(20\) feet you earn \(40\) points, and so on.

Rational Functions

Rational functions are formed by dividing one polynomial by another. Like polynomials, rational functions can be written in more than one way. Remember when we multiplied \((x^2 + 3)(x^2 - x -1)\) and used the rules of exponents to find that their product was another polynomial whose highest exponent was the sum of the two exponents we started with? When we perform a process known as polynomial long division on a rational function, we are essentially trying to reverse this process.

Let’s divide \(x^3 - 7x + 6\) by \(x-2\). Since the dividend does not have an \(x^2\) term, when setting up the long division, make sure to write \(0x^2\). It makes the division process easier. Polynomial long division is essentially the same as normal number division. The long division and quotient (answer) is shown below:

First, we divide \(x^3\) by \(x\) to get \(x^2\). We write it above. Then, multiply \(x\) by \(x^2\) (make sure to flip the sign) and write \(-x^3\) below. It cancels out. Then, you multiply \(-2\) by \(x^2\) (flipping the sign again) and write \(2x^2\) below. Adding, we get \(2x^2\) in the line below and bring down \(-7x\). Then we repeat the process with \(-7x\) being divided by \(x-2\), with individual terms. We stop until we get \(0\).

Thus, \(x^3 - 7x + 6\) divided by \(x-2\) gives us:

\[x^2 + 2x-3\]

You could now multiply \(x-2\) by \(x^2 + 2x - 3\) to check if your division is correct or not.

Function Representation Relationships

In some of these questions, you will rely heavily on the ability to understand and coordinate graphic and algebraic representations of the same function. Be sure you understand the roles of these components in this type of math.

Domain and Range

The domain is the set of values that a function can take as an argument (i.e., input), while the range is the set of all values the function can give as the output. On a graph, the domain would be the set of all \(x\)-coordinates that are points on the graph, and the range is the set of all \(y\)-coordinates that are points on the graph.

Minimum and Maximum Values

Maximum and minimum values are exactly what their names say: the highest and lowest values a function attains, respectively. On a graph, they’re easy to identify: The maximum is the highest a function goes, and the minimum is the lowest it goes. Just be careful not to confuse the function’s maximum value, which is the \(y\)-coordinate of the highest point, with where it reaches the maximum, which is the \(x\)-coordinate. The same goes for the minimum. Of course, functions may not have one or either of them. Many functions just keep increasing or decreasing.

Increasing and Decreasing

Increasing and decreasing describe how the function’s value changes as its argument changes. If the function’s value grows larger as its argument grows larger, the function is increasing, and if the function’s value becomes smaller as its argument grows larger, the function is decreasing. (When values are negative, larger can also mean “more positive” and smaller can mean “more negative”). Graphically, the function is increasing if the line or curve representing it goes up as it moves to the right, and the function is decreasing if the line or curve goes downward.

Functions can be increasing in some areas and decreasing in others. For example, the function below is increasing until \(x = 0\), then decreasing from \(x = 0\) to \(x = 2\), and increasing thereafter.

End Behavior

End behavior refers to how a function behaves as its argument increases or decreases as far as we can take it. In a graph, this is what the function does toward either end, looking from left to right. For example, the function below approaches infinity as its argument grows larger, and it approaches zero as the argument becomes smaller.

Asymptotes

Asymptotes are lines that functions approach but never quite touch. In the example below, the line \(y = 0\) is an asymptote. To the naked eye, it might not be entirely clear, but the function never actually reaches \(0\) and never would even if we were to follow the function forever.

More generally, the asymptote is \(y = b\), where \(b\) is the value \(y\) approaches. Asymptotes can be horizontal, vertical, or even slanted. Below is a function (in red) with a slanted asymptote and a vertical asymptote (in black). The slanted asymptote has the equation \(y = x - 3\). A rational function’s vertical asymptotes can be found by looking for where its denominator is \(0\). As the denominator decreases, the value of the fraction increases. For the function below, we see:

\[x + 3 = 0\]

Therefore, \(x = - 3\) is the vertical asymptote.

Symmetry

Sometimes a function will look like a mirror image. In mathematical terms, we call a function symmetric about an axis if its value at any distance to one side of the axis is the same as the value at the same distance to the other side.

In function notation, \(f\) is symmetric with respect to the axis \(x = a\) if \(f(x - a) = f(-(x - a))\).

\[f(x) = \frac{1}{3}x^4 - \frac{10}{3}x^2 + 3\]

The function (in red) above is symmetric with respect to the line \(x = 0\) (in orange).

Transformations

Suppose we have a function described as \(f(x)\). Since the horizontal axis is associated with a function’s input, \(x\), we can move everything by shifting the input: Changing \(f(x)\) to \(f(x+a)\) moves the graph horizontally left by \(a\) units. Likewise, since the vertical axis is associated with the output, \(f(x)\), we can shift the function vertically up \(a\) units by changing \(f(x)\) to \(f(x) + a\).

Be careful with the direction of a horizontal shift as the graph will move in the direction opposite to the sign of \(a\). For example, the value of \(f(x - 2)\) when \(x = 3\) is the same as the value of \(f(x)\) when \(x = 1\), so a negative number means a positive shift in the input. After all, \(x\) has to increase to “balance out” the negative shift from \(-2\).

Note: This is only for horizontal shifts. For a vertical shift, the direction of the shift will move in the same direction as the sign of \(a\).

Creating a Function

On the SAT, you may encounter an item that requires you to create a function for a word problem. It will describe a real-world situation and then ask you to create a function that models it. In these items, the key is to understand the relationship between the variables and express that relationship mathematically. Often the problem will describe how one quantity changes when another quantity increases or decreases. Your job is to translate that relationship into a function rule. Let’s look at an example.

Example

A small concert venue tracks how the number of tickets sold affects the average ticket price. The manager notices that as more tickets are available, the average price decreases. Let \(p\) be the average ticket price (in dollars) and \(n\) be the number of tickets available.

Which of the following could represent \(p\) as a function of \(n\)?

A. \(p(n) = 12n\)

B. \(p(n) = \frac{300}{n}\)

C. \(p(n) = n - 25\)

D. \(p(n) = \frac{n}{15}\)

Solution

The question tells us that when the venue has more tickets available, the average ticket price goes down. That means the two variables move in opposite directions. When one increases, the other decreases. This kind of relationship is called an inverse relationship.

Now look at the answer choices and think about what each one does.

Answer choice A involves simple multiplication, meaning the price increase as more tickets are available. That’s the opposite of what the problem describes.

In answer choice C, despite the subtraction, the price still increases when \(n\) increases, so this can’t be right.

Likewise, answer choice D gets bigger as \(n\) gets bigger, so this isn’t right.

In answer choice B, \(n\) is in the denominator, so as \(n\) gets larger, the value of the function (ticket price) gets smaller. For example, when \(n\) is \(50\), the ticket price is \(300 \div 50 = 6\), and when \(n\) is \(100\), the ticket price is \(300 \div 100 = 3\). So, the correct answer is B.