Upper Level: Quantitative Reasoning Study Guide for the ISEE

Algebra

Basic Algebraic Notation

The basic notations used in arithmetic are carried over to algebra, such as the symbols for the operations: \(+\), \(-\), \(\cdot\), and / or ÷. Grouping symbols include the parentheses and brackets (square and curly brackets). Symbols that compare values are:

\(=\) for equality
\(\sim\) and \(\simeq\) for equivalence relations
\(\neq\), \(\ge\), \(\le\), \(\gt\), and \(\lt\) for inequality

The polarity, or signs used before numbers or variables, could be \(+\), \(-\) or \(\pm\). The absence of a sign indicates that the number or variable is positive.

Factorials are represented by !, and radicals by a fractional exponent or this symbol: \(\sqrt{}\).

You must be familiar with other symbols and their values or meanings, as well, including \(\pi\), \(\infty\), \(\vert\;\vert\), and many others.

Function notations and the function composition symbol are unique to functions, and they will be taken up under the next two headings.

Polynomials

Polynomials are algebraic expressions with more than one term containing constants, variables, or exponents, except division by a variable.

A question may go this way:

Which of the following is not a polynomial?

(A) \(\sqrt {6}\)

(B) \(2x^3 – x^{-2} + 5\)

(C) \(x^2 + 6x + 9\)

(D) \((a + 4b)^2\)

Recall that a polynomial is an algebraic expression that may contain numbers, variables, or variables with exponents. However, it can never have division by a variable.

Choice A is simply a number, \(\sqrt{6} \approx 2.4495\), so it is a polynomial of the most basic form.

Choice C is also a polynomial; it consists of numbers and variables with exponents. It is a three-term polynomial called a trinomial.

Choice D is a polynomial as well. The expanded version of this polynomial is:

\[(a+4b)^{2}\] \[= (a)^{2} + 2(a)(4b) + (4b)^{2}\] \[= a^{2} + 8ab + 16b^{2}\]

This is an expression with numbers and variable exponents, so it is a polynomial.

The correct answer is (B).

Choice B is not a polynomial because it has division by a variable. Recall the exponent property \(x^{-a} = \frac{1}{x^a}\). We can use this exponent property with the second term of the expression:

\[2x^3 - x^{-2} +5\] \[= 2x^3 -\frac{1}{x^2} + 5\]

As you can see, the second term includes division by a variable, so the expression is not a polynomial.

Equations

Practice adequately with different types of equations. Familiarize yourself with linear equations, the standard equation of a line, point-slope form, slope-intercept form, determining the midpoint of a line segment, and the relationship between parallel and perpendicular lines.

Many questions will involve second-degree equations, so make sure you have the skills for factoring quadratic equations, completing the square, using the quadratic formula. A lot of problems will involve your ability to visualize these equations, so make sure you know how to graph them as well.

Functions

An equation or algebraic relationship is considered a function if, for every input value, it produces exactly one output value. This is called the vertical line test and all functions must satisfy this condition. While functions are commonly written in the \(f(x)\) form, the following are just some examples of functions:

\[h(x) = 3x^2\] \[g(x) = \frac{2}{5} x + 6\]

In fact, it’s not always \(x\); it can be \(y\), \(\theta\), or other symbols.

Observe the following rules of operating with functions:

To add functions: \((h+g)(x) = h(x) + g(x)\)

To subtract functions: \((h-g)(x) = h(x) - g(x)\)

To multiply functions: \((h·g)(x) = h(x) · g(x)\)

To divide functions: \(\frac{h}{g})(x) = \frac{h(x)}{g(x)}\)

Composition of Functions

This function notation \((h \cdot g)(x)\) refers to composition, or application of the function \(h(x)\), which results in the function \(g(x)\).

To perform function composition: \((h \cdot g)(x) = h[g(x)]\)

Let’s do this for the sample functions given above:

\[(h \cdot g)(x) = 3(\frac{2}{5}x + 6)^2\]

Take note that this is different from the result we get for this composition:

\[(g \cdot h)(x) = \frac{2}{5}(3x^2) + 6\]

Be sure to practice evaluating expressions with function notations, such as in this question:

Compare the quantities in Column A and Column B if \(f(x) = x+4\) and \(g(x) = \frac{2}{3}x\).

\[\begin{array}{c|c} \text{Column A} & \text{Column B} \\ \hline f(2) - 4 & g(3) \\ \end{array}\]

In this type of question, we have to compare the quantities in Column A and Column B. The answer to this question will resemble this format:

Answer “A” if the quantity in Column A is greater. Answer “B” if the quantity in Column B is greater. Answer “C” if the two quantities are equal. Answer “D” if the comparison cannot be determined from the information given.

Let’s evaluate the function in Column A:

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

Now, let’s evaluate the function in Column B:

\[g(x) = \frac{2}{3}x\] \[g(3) = \left(\frac{2}{3}\right)(3) = 2\]

Since both the quantities in Column A and B are equal, the correct multiple choice answer is “C”.

Analyzing Functions of One Variable

Linear functions are functions of one variable, such as this one below:

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

When \(x\) changes, \(f(x)\) changes at a constant rate. In a line, this rate of change (\(m\)) also called the slope.

When graphing linear functions, we need to know the intercept of the function when \(x\) is zero. In the function above, the graph crosses the \(y\)-axis at \(b\), also called the \(y\)-intercept.

Functions of one variable may have degrees higher than one, such as this parabolic function:

\[f(x) = x^2 – 16\]

The roots or zeroes of this function are the values of \(x\) when \(f(x)\) is equal to zero. Solving for the roots, we find \(x = +4\) and \(x = -4\).

\[f(x) = x^2 -16\] \[0 = x^2 -16\] \[x^2 = 16\] \[\sqrt{x^2} = \sqrt{16}\] \[x = \pm 4\]

Additional readings may include functions in one variable but of degrees higher than \(2\).

Understanding Representations of Functions of Two Variables

Expanding on the concept of functions in one variable, here’s how you may visualize functions of two variables. Using the conventional \(x\)- and \(y\)-axes, add to your visualization the \(z\)-axis which is perpendicular to both axes. The \(xyz\)-axes should create a \(3\)-dimensional space in your mind (or your graph), and a function of two variables creates surfaces occupying that space.

Thus, this function actually involves \(3\) variables and creates a surface in a \(3\)-dimensional space.

\[f(x, y) = 5 – x + 3y\]

For ease in graphing or visualization, this function notation may also be written as

\[z = 5 – x + 3y\]

A question may ask, for instance, for the points of intersection of \(f(x, y)\), as defined in the above sample function, with the \(xyz\)-axes.

The point where the graph of the equation cuts the \(x\)-axis can be found by setting both \(y\) and \(z\) to \(0\):

\[z = 5 - x + 3y\] \[0 = 5 - x + 3(0)\] \[0 = 5 - x\] \[x = 5\]

Thus, the \(x\)-intercept is at \((5,0,0)\).

The point where the graph of the equation cuts the \(y\)-axis can be found by setting both \(x\) and \(z\) to \(0\):

\[z = 5 - x + 3y\] \[0 = 5 - (0) + 3y\] \[0 = 5 + 3y\] \[3y = -5\] \[y = -\frac{5}{3}\]

Thus, the \(y\)-intercept is at \(\left(0,-\frac{5}{3},0\right)\).

The point where the graph of the equation cuts the \(z\)-axis can be found by setting both \(x\) and \(y\) to \(0\):

\[z = 5 - x + 3y\] \[z = 5 - (0) + 3(0)\] \[z = 5\]

Thus, the \(z\)-intercept is at \((0,0,5)\).

Your answer should be the choice that gives \((0, 0, 5), \;(5, 0, 0),\) and \((0, -\frac{5}{3}, 0)\). This is an example of a function involving an equation of a plane.

Take note that the domain of a function of two variables consists of ordered pairs \((x, y)\) from the \(xy\) space and which can be inputted to the function to get the output value for \(z\).

Inequalities

To deal with inequalities in algebra, you have to know the following properties of inequalities:

Transitive property: If \(x \gt y \gt z\), then \(x \gt z\)

Reversal property: If \(x \gt y\), then \(y \lt x\)

Non-negative property of squares: \(x^2 \ge 0\)

Square root property: If \(x \ge y\), then \(\sqrt{x} \ge \sqrt{y}\) for \(x \ge 0\) and \(y \ge 0\)

Rule for solving inequalities: When a negative value is added or subtracted to both sides of an inequality, this will not change the direction of the inequality sign. However, when a negative value is multiplied or divided to both sides of an inequality, the inequality sign changes in direction. The sign also changes in direction when the left and right sides of the inequality are swapped.