Upper Level: Mathematics Achievement Study Guide for the ISEE

Algebra

Patterns, Relationships, and Functions

The Mathematics Achievement section will test your understanding of functions and relationships, and generalize patterns or graphical representations for them.

Relationships are generally expressed in equations or inequalities. A mathematical relationship that generates exactly one output value for every input value is a function. Because of this definition (which is also known as the vertical line test), not all equations satisfy this requirement and qualify as functions.

Test your ability to visualize or plot various types or classes of functions. To do this, learn about intercepts, roots of a function, slope of a line, vertices, degree of a function, locating points on the Cartesian axes, and the approximate shape of a function.

You may be asked to find the zeroes of a function. Another question may ask for the coordinate points where two given functions are true. Find more examples and drills on this.

When performing basic operations with functions, observe these rules:

Addition: $(h + g)(x) = h(x) + g(x)$

Subtraction: $(h - g)(x) = h(x) - g(x)$

Multiplication: $(h \cdot g)(x) = h(x) \cdot g(x)$

Division: $(\frac{h}{g})(x) = \frac{h(x)}{g(x)}$

Analyzing Functions

Visualizing functions is one way to analyze functions. Given a function of one variable, you should know how the function behaves as the variable changes. A degree-one function, for example, is given below as:

\[f(x) = 2x - 5\]

When the variable $x$ is zero, *$f(x)$ is $-5$. When $x = 3$, $f(x)$ is $1$.

In fact, in this example, $f(x)$ changes at the rate of $2$ times the value of $x$. In linear functions, this rate of change is also known as the slope, which is further defined as the change in rise over the change in run.

Conversely, you must also recognize functions, and possibly interpret them numerically, when they are presented in the graphical form.

For example, a parabola opening upward, with vertex at the origin $(0, 0)$, and plotted with these values:

\[\begin{array}{c|c} \text{x} & \text{y} \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \\ 4 & 16 \\ \end{array}\]

### Function Transformation

Functions can be transformed—moved around the $xy$-axes and resized.

The function $g(x) = x^2 + 4$ is a similar function as the parabola defined above, except that its vertex is moved or slid along the $y$-axis $4$ spaces up ($+4$).

The function $h(x) = x^2 – \;9$ is a transformation of $f(x) = x^2$ with its vertex at point $(0, -9)$. It will cross the $x$-axis at $(-3, 0)$ and $(3, 0)$.

Function $f(x)$ can be moved to the left or to the right by adding a positive or negative constant to $x$. Moving $f(x)$ $2$ spaces to the left can be written as:

\[f(x) = (x+2)^2\]

The rate of change is determined by the coefficient of $x$. When this coefficient is positive, the function faces upward. When the coefficient is negative, the function faces downward. A coefficient bigger than $1$ compresses the function; a coefficient less than $1$ but more than $0$ stretches the function.

Functions with One and Two Variables

Functions may have one or more variables.

A function of one variable operates on the $xy$-plane. The examples given above, so far, were of this type. A linear function changes at a constant rate along the $y$-axis as the value of $x$ changes along the $x$-axis. The domain of the function consists of the values for $x$ which when inputted to the function will produce the value for $f(x)$ or $y$ in our graph.

A function of two variables operates on a $3$-dimensional $xyz$-space. Expanding from the $xy$-plane, add a third axis $z$, which is perpendicular to both $x$ and $y$ axes. Functions of two variables create surfaces occupying this space. It must be noted that the domain of functions such as this consists of ordered pairs for $(x, y)$ which when inputted to the function will produce the value for $f(x, y)$ or $z$ in our graph.

Classes and Properties of Functions

Familiarize yourself with the different types of functions and their properties. Here are some of the functions commonly encountered in the ISEE test.

Linear function— this function is a straight line with its standard form written as:

\[f(x) = mx + b\]

where $m$ is the slope of the line (change in rise over change in run) and $b$ is the $y$-intercept

Square function (or quadratic function)— this function can be written in the general form:

\[f(x) = Ax^2 + Bx + C\]

where $A$, $B$ and $C$ are real numbers, and $A$ ≠ $0$

The domain of this function is all real numbers, and its range is all positive real numbers, including zero, when the $y$-intercept is $0$ and $A \lt 0$.

It may also be written in the form:

\[f(x) = A(x-h)^2 + k\]

where ($h$, $k$) is the vertex and $A$ defines the width or spread of the parabola

Cube function— in general form, this function is written as:

\[f(x) = Ax^3 + Bx^2 + Cx + D\]

where $A$, $B$, $C$ and $D$ are real numbers, and $A$ ≠ $0$

The domain of the function is all real numbers; the range is all real numbers.

Square root function— this is a function written as:

\[f(x) = \pm \sqrt {x-h} + k\]

The domain is all positive real numbers; the range is all positive real numbers.

Absolute value function— an absolute value function with vertex at ($h$, $k$) will be written as:

\[f(x) = A \mid x – h \mid + k\]

$A$ defines how widely spread the graph of the function will be; $h$ moves the function $h$ spaces to left or right of the origin, while $k$ moves the function $k$ spaces up or down from the origin.

Read further on reciprocal, floor-and-ceiling, exponential and trigonometric functions, and end behavior of a function.

Equalities and Inequalities

Mathematical statements are used to express equality and inequality relationships. They often involve one or more variables and may be in degree one or higher.

A linear equation involves two variables, and it can be written several ways. A line that passes through points $(-3, 0)$ and $(0, 2)$ can be written in any of the following forms:

Standard form: $Ax + Bx + C = 0$ : $2x – 3y + 6 = 0$

Slope-intercept form: $y = mx + b$ : $y = \frac{2}{3}x + 2$

Point-slope form: $y – y_1 = m(x – x_1)$ : $y-0 = \frac{2}{3}(x +3)$

Plotting this equation will yield a line sloping upward, crossing the $x$-intercept at $(-3, 0)$ and the $y$-intercept at $(0, 2)$.

Let’s consider linear inequalities. For example, a very similar inequality to the one given above is written as:

\[2x -3y + 6 < 0\]

This linear inequality will be plotted similarly, but with a broken line and a shaded area either above or below the broken line. The shaded area represents the solution set to this inequality. To know which part to shade, find a random point above the line, say $(-1, 3)$ and plug the $x$ and $y$ values of this point to the inequality to see if it will yield a true statement:

\[2(-1) – 3(3) + 6 < 0\] \[-2 -9 + 6 < 0\] \[-5 < 0\]

This is a true statement, and this confirms that point $(-1, 3)$, which is above the line, is part of the solution set. Shading the area above the line indicates that all points in the shaded area are part of the solution set.

Study and find additional resources on inequalities involving $\gt$, $\le$, and $\ge$, the rules of operating with inequalities, and equalities and inequalities in degrees higher than $1$, such as quadratic equations.

Using Symbolic Representation

Algebra deals with signed numbers, called integers, so familiarize yourself with the common algebraic symbols and notations used. Variables, such as $x$, $y$, $a$, $b$, and other letters represent unknown values. These signs “$+, \;-,\; and$\pm$$” are written before numbers and variables to show polarity, but a number that carries no sign is understood to be positive.

Mathematical statements or expressions are written in a mathematical language using numbers or constants, variables, exponents and coefficients. Expressions that are equal are linked in equations with the symbol of equality ($=$) in the middle. Expressions that are not equal are written similarly but with appropriate symbols defining their relationship, such as $\gt$, $\lt$, $ge$, $le$ or $neq$.

Other symbols used are “!” (called factorials), “$\sqrt { }$” (for radicals), and “$\mid \mid$” for absolute value. Some symbols have assigned values, such as $\pi$ for pi which has a value of approximately $3.14159$.

Solving Equations: The Basics

Questions in this section will mainly involve solving equations. Equations may be given directly, and you will be asked to solve for an unknown variable, or you will be given a word problem for which you will need to write your own equation and solve for what’s being asked. Either way, the ability to understand and evaluate equations is important.

Remember always that the goal of equations is to solve for the unknown using known values and relationships. In an equation with one variable, you only need to isolate the variable to one side and evaluate the numbers on the other side.

For instance, you are asked to solve for $x$ in this equation:

\[x-5 = 1-2x\]

Isolating $x$:

\[x+2x = 1+5\] \[3x = 6\] \[x = 2\]

That was easy; solving equations involving several variables will be more challenging. Always remember, when solving for three unknowns, you need at least $3$ equations.

Equations are also seen as formulas, such as “$A = \frac{1}{2}$base $\times$ height*”.

Find additional materials to review on systems of equations, solving the unknown by the methods of elimination and substitution, graphing, and solving equations using matrices.

Using Graphical Data

Graphing data is another way of solving a problem, an equation or a function. It helps you visualize and reason mathematically and find points on a curve. Plotting or graphing makes use of the Cartesian axes, or $xy$-axes, and sometimes the $xyz$ space, location of points relative to these axes, and the known values of the given equations. The solution, for instance, to a system of two linear equations is the point of intersection of these two lines.

If you graph these two lines:

Line 1: $2x - y -1 = 0$

Line 2: $y = 2 – x$

you will find their intersection at point $(1, 1)$. This is the solution to this system of equations.

Lines of the same slope (parallel lines) but different $y$-intercepts will have no solution. Refresh your skill in plotting graphs of quadratic equations.