Math Study Guide for the SAT Exam

More Algebraic Skills to Practice

Determine a Solution Set

Determining a solution set is equivalent to finding all values that will satisfy an equation or inequality. It is worthwhile to recognize the relationship between the number and type of variable or variables present in an equation or inequality. It is also valuable to understand the solution sets possible for systems of equations.

Consider the following equations:

$x + 1 = 2$ and $x + 1 = x$

The solution set for the first is $\{1\}$ because only when $x = 1$ is the equation true. The solution set for the second is the null set, meaning that there is no value that will make the equation true—there is no solution.

In the case of an equation containing two first-degree variables, such as $(y = mx)$, the solution set will be multiple points that lie in a straight line. Contrastingly, the solution set of a linear inequality will either be the collection of points above or below a line, with the points along the line also being part of the solution depending on whether the inequality contains an equal sign.

A solution set can always be verified by substituting values back into the original expression to determine whether the equation produces a true statement.

Determine if a System of Equations Represents a Context

Problems that present two distinct pieces of information that can be represented with variables are prime candidates for solving by way of a system of equations. Consider the following problem:

Pens cost $50$ cents and pencils cost $25$ cents. Amanda purchased a total of $9$ pens and pencils for a cost of $\$4.25$. What is the number of pens she bought and what is the number of pencils she bought?

This problem presents two pieces of information—the total number of pens and pencils and the prices of each. Both the associated costs and the total amount are fixed values, but the number of pens and pencils are both unknown. Let’s use $x$ to represent the number of pens and $y$ to represent the number of pencils. We can generate a system of equations as follows:

\[50x + 25y = 425\] \[x + y = 9\]

Solving the second equation for $y$ and substituting the expression into the first equation:

\[y = 9 - x\] \[50x + 25(9 - x) = 425\] \[50x + 225 - 25x = 425\] \[25x = 200\] \[x = 8\]

Substituting this value into the second equation:

\[8 + y = 9\] \[y = 1\]

Amanda bought $8$ pens and $1$ pencil.

Determine the Meaning of All Terms in an Equation or Expression

In other words, what do they stand for in terms of the context given in the problem or question? Let’s analyze the previous problem statement and its corresponding system of equations.

Pens cost $50$ cents, and pencils cost $25$ cents. Amanda purchased a total of $9$ pens and pencils for a cost of $\$4.25$. What is the number of pens she bought, and what is the number of pencils she bought?

The system of equations are:

\[50x + 25y = 425\] \[x + y = 9\]

The first equation corresponds to the associated cost of the pens and pencils. Notice that the original problem statement presents the information in terms of dollars and cents. However, by multiplying each dollar amount by $100$, we can eliminate the decimals and make the numbers easier to work with. Notice that the coefficient of the $x$, which we assumed to be the unknown number of pens is $50$—this is because we were given that the cost of every pen was $50$ cents (and multiplying by $100$ gives $50$). Likewise, the coefficient of the $y$, which represents the unknown number of pencils, is $25$.

The product of the unknown number of pens with the cost per pen gives the total cost of the pens, and the product of the unknown number of pencils with the cost per pencil gives the total cost of the pencils. This combined cost represents the total cost of pens and pencils, which is given in the problem statement.

The problem statement also gives us the total number of pens and pencils purchased. Because $x$ represents the unknown number of pens and $y$ represents the unknown number of pencils, $x + y$ is equivalent to the total number of pens and pencils.

By understanding which elements of a problem represent constants and which elements represent variables, it is manageable to generate an equation or system of equations that models the situation.

Match a Given Linear Equation to Its Corresponding Graph

Linear equations provide information about the slope and intercept(s) of a particular line.

The slope is the ratio of the change in $y$ values to the change in $x$ values, often written as $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are points on the line. The slope is commonly remembered by the expression rise over run.
The intercepts of a line are the points at which a line crosses the $x$ or $y$ axis. These values can be determined by setting the $x$ or $y$ variable equal to $0$ and solving for the remaining variable.

To match a given linear equation with its corresponding graph, begin by finding the slope and intercept(s) of the equation, and then find the graph that exhibits the same slope and intercept(s).

Match a Given Graph to Its Corresponding Equation

From a provided graph, you can determine both the slope and any axes intercepts present. In order to match a given graph with its corresponding equation, begin by selecting two points along the line and then determine the ratio of the change in the $y$ values to the change in the $x$ values.

Next, find the $y$-intercept of the line (in the case where no $y$-intercept is present, then the line is vertical and contains only one variable, $x$).

From these two values, the slope-intercept form of the line can be written: $y = mx + b$, where $m$ is the slope, and $b$ is the $y$-intercept. If the equation generated does not match an available answer choice, it may be necessary to algebraically manipulate the equation so that it does match an answer choice.

Match a Verbal Graph Description to Its Corresponding Equation

Occasionally you will be asked to match the description of a graph with its corresponding equation. For situations like this, it is important to understand common terminology, such as, “The following data exhibits a linear/exponential/inverse, etc. relationship…”

As a result, familiarity with these terms is important.

The $x$ or $y$ intercept of a function is the point at which the function crosses the $x$ or $y$ axis, respectively. These points are located by evaluating the function with $x$ or $y$ equal to $0$. When $x$ is equal to $0$, you are solving for the $y$-intercept. When $y$ is equal to $0$, you are solving for the $x$-intercept.
Linear equations produce straight-line graphs. Lines can be positive and slope upward as the $x$ value increases, or they can be negative and slope downward as the $x$ value increases.
Quadratic equations produce parabolas. Parabolas can be positive, have a vertex as an absolute minimum, and extend upwards to infinity; or, they can be negative, have a vertex as an absolute maximum, and extend downwards to infinity. Quadratic equations will always have a $y$-intercept, but there are different cases regarding their $x$-intercepts. A quadratic that has only imaginary solutions does not cross the $x$-axis. A quadratic that has only one real solution has its vertex along the $x$-axis. A quadratic that has two solutions crosses the $x$-axis at two points. The $x$ intercepts of a quadratic are also called the zeroes of the quadratic.
Cubic equations can be positive or negative. As the $x$ value increases, positive cubic functions extend from negative infinity toward a point of inflection before curving upward to positive infinity. Negative cubic functions exhibit the opposite—as the $x$ value increases, they extend from positive infinity to a point of inflection before curving downward to negative infinity.

Determine Graph Features by Examining Its Equation

Develop a familiarity with each of these general parent functions:

$x = a$ where $a$ is a constant

This is a vertical line.

$y = a$ where $a$ is a constant

This is a horizontal line.

$y = ax$, $y > ax$, $y < ax$, $y \le ax$, $y \ge ax$ where $a$ is a constant

These all graph as a solid or dotted line with a slope of $a$.

$y = \vert ax \vert$ (absolute value of $ax$)

This produces two lines with opposite slopes that emanate from a single point.

$y = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants

This graphs as a parabola. It is also known as a quadratic function and an even function. All even functions exhibit end behavior that matches in sign- both ends extend in the same direction.

$y = ax^3 + bx^2 + cx + d$ where $a$, $b$, $c$, and $d$ are constants

This produces a graph that extends from negative infinity through a point of inflection to positive infinity. It is like starting with a quadratic but inverting one-half of it to the left or right of the vertex. This is known as a cubic, or odd, function. All odd functions exhibit end behavior that is opposite in direction—both ends extend in opposite directions.

Describe Graph Changes When an Equation is Changed

Familiarity with parent functions and the general rules dictating horizontal and vertical translations, as well as dilations and scaling factors, enables a quick assessment of how a function will graph.

Consider the parent function of a linear equation: $y = x$

This plots as a line with a slope of $1$. But, if we add a constant to either side of the equation, the line will vertically shift either up or down, depending on the sign of the constant.

Multiplying the $x$ variable by a constant will produce another type of change in the line’s graph. If we multiply it by a negative $1$, the slope of the line becomes negative, meaning as the $x$ values increase, the $y$ values decrease. Multiplying the $x$ variable by a large positive constant increases the steepness of the line, whereas multiplying it by a very small positive constant decreases the steepness of the line.

Similar translations take place when other functions are manipulated in the same manner. You should become familiar with whether a function will shift horizontally or vertically, as well as whether a function will be stretched or dilated.

Some general trends can be deduced from a specific example. Consider the following:

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

This should be recognized as the vertex form of a parabola. Usually, a parabola has its vertex at the origin, $(0,0)$, but in this case, because of the negative sign in front of the parentheses, the subtraction of $4$ inside the parentheses, and the addition of $3$, the vertex of the parabola will be shifted.

The addition or subtraction of a constant from the $x$ variable produces a horizontal shift. In this case, the $x$ value of the vertex will be shifted $4$ places to the right. If instead, the parentheses contained $(x + 4)$, the $x$ value of the vertex would be shifted $4$ places to the left. The shift is in the opposite direction of the sign. The addition of $3$ on the outside results in an upward vertical translation. If, instead, there was a $-3$, the vertical translation would be downward. The negative sign in front of the parentheses produces an inversion—rather than the parabola extending upward and possessing an absolute minimum, the parabola extends downward and possesses an absolute maximum.