ACCUPLACER Advanced Algebra and Functions: Precalculus Study Notes

Linear Equations and Functions

Definition and Properties

Linear equations and functions are foundational concepts in precalculus, describing relationships with constant rates of change. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

• Slope (m): Measures the steepness of the line; calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Y-intercept (b): The point where the line crosses the y-axis.

• Perpendicular Lines: Two lines are perpendicular if their slopes are negative reciprocals.

Example: Find the equation of a line passing through (0,4) and perpendicular to $y = -\frac{3}{2}x + 2$. The slope of the perpendicular line is $\frac{2}{3}$, so the equation is $y = \frac{2}{3}x + 4$.

Factoring and Quadratic Equations

Quadratic Functions and Their Graphs

Quadratic equations are polynomials of degree two, typically written as $y = ax^2 + bx + c$. Their graphs are parabolas.

• Vertex: The highest or lowest point of the parabola, given by $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line passing through the vertex.

• Factoring: Expressing a quadratic as a product of two binomials, e.g., $x^2 + 5x + 6 = (x+2)(x+3)$.

Example: The graph of $y = f(x)$ is shown as a downward-opening parabola. If the vertex is at (2, -4), possible equations include $f(x) = -(x-2)^2 - 4$.

Functions and Their Properties

Evaluating and Transforming Functions

A function assigns each input exactly one output. Functions can be evaluated, transformed, and combined.

• Function Notation: $f(x)$ denotes the output when input is $x$.

• Evaluating Functions: Substitute the input value into the function rule.

• Transformations: Shifting, stretching, or reflecting the graph of a function.

Example: If $g(x) = 5(x + 8)$, then $g(12) = 5(12 + 8) = 100$.

Radical and Rational Expressions

Simplifying and Solving

Radical expressions involve roots, while rational expressions are ratios of polynomials.

• Simplifying Radicals: $\sqrt{a^2} = a$ if $a \geq 0$.

• Rational Expressions: Simplify by factoring numerator and denominator and reducing common factors.

• Operations: Addition, subtraction, multiplication, and division follow algebraic rules.

Example: $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$.

Polynomial and Exponential Equations

Solving and Graphing

Polynomials are expressions with terms of the form $ax^n$. Exponential equations have variables in the exponent.

• Polynomial Equations: Set equal to zero and solve for roots.

• Exponential Equations: Use logarithms to solve for the variable.

• Graphing: Recognize shapes of polynomial and exponential graphs.

Example: $500(2)^t$ gives the expected number of bacteria after $t$ days.

Logarithmic Expressions

Properties and Simplification

Logarithms are the inverses of exponentials. The logarithm base $b$ of $x$ is the exponent to which $b$ must be raised to get $x$.

• Product Rule: $\log_b(xy) = \log_b x + \log_b y$

• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

• Power Rule: $\log_b(x^k) = k \log_b x$

Example: $\log_2 8 = 3$ because $2^3 = 8$.

Geometry Concepts

Triangles and Area

Geometry in precalculus includes properties of triangles, area calculations, and relationships between angles and sides.

• Area of Rectangle: $A = \text{length} \times \text{width}$

• Congruent Triangles: Triangles are congruent if corresponding sides and angles are equal.

• Pythagorean Theorem: In a right triangle, $a^2 + b^2 = c^2$.

Example: In triangle ABC, if angle C is a right angle and $\sin A = \frac{3}{5}$, then $A$ is an angle in a right triangle with opposite side 3 and hypotenuse 5.

Trigonometry

Basic Trigonometric Functions

Trigonometry studies relationships between angles and sides in triangles. The primary functions are sine, cosine, and tangent.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Example: If $\sin A = \frac{3}{5}$, then the triangle sides can be 3 (opposite), 4 (adjacent), and 5 (hypotenuse).

Sample Table: Cost Comparison of Apples and Pears

The following table compares the cost per pound of apples and pears based on the number of pounds purchased.

Number of Pounds	Cost of Apples ($)	Cost of Pears ($)
1	Approx. 0.50	Approx. 0.57
2	Approx. 1.00	Approx. 1.14
3	Approx. 1.50	Approx. 1.71

Additional info: Table values inferred from graph and question context.

Additional Info

Additional info: These notes are based on sample questions from the ACCUPLACER Advanced Algebra and Functions test, which covers precalculus topics including linear equations, factoring, functions, quadratics, radicals, polynomials, exponentials, logarithms, geometry, and trigonometry.