College Algebra: Midterm Exam 3 – Sample Questions Study Guide

Polynomial Functions and Equations

Solving Polynomial Equations

Polynomial equations are equations involving expressions with variables raised to whole number powers. Solving these equations often involves factoring, using the Rational Root Theorem, or applying synthetic division.

• Rational Root Theorem: This theorem helps to find possible rational roots of a polynomial equation with integer coefficients. If $f(x) = a_nx^n + ... + a_0$, then any rational root $\frac{p}{q}$ must have $p$ dividing $a_0$ and $q$ dividing $a_n$.

• Factoring: Expressing a polynomial as a product of its factors can simplify solving $f(x) = 0$.

• Synthetic Division: A shortcut method for dividing a polynomial by a linear factor of the form $(x - c)$.

• Example: Solve $f(x) = x^3 + 8x^2 + 11x - 20 = 0$ using the Rational Root Theorem to list possible rational roots, then test them by substitution or synthetic division.

Graphing Polynomial Functions

Graphing polynomials involves identifying intercepts, end behavior, and turning points.

• x-intercepts: Values of $x$ where $f(x) = 0$.

• y-intercept: Value of $f(0)$.

• End Behavior: Determined by the leading term $a_nx^n$.

• Turning Points: A polynomial of degree $n$ can have up to $n-1$ turning points.

• Example: Sketch the graph of $f(x) = x^3 + 8x^2 + 11x - 20$ by finding intercepts and analyzing end behavior.

Rational Functions

Properties and Graphing

Rational functions are quotients of polynomials. Their graphs can have vertical and horizontal asymptotes, holes, and intercepts.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator.

• Holes: Occur where a factor cancels from numerator and denominator.

• Example: For $g(x) = \frac{8x}{x^2 - 1}$, factor denominator as $(x-1)(x+1)$. Vertical asymptotes at $x=1$ and $x=-1$; horizontal asymptote at $y=0$.

Functions and Their Properties

Function Operations and Composition

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Sum, Difference, Product, Quotient: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x)g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$).

• Composition: $(f \circ g)(x) = f(g(x))$.

• Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Domain of Functions

The domain of a function is the set of all input values for which the function is defined.

• Polynomial Functions: Domain is all real numbers.

• Rational Functions: Exclude values that make the denominator zero.

• Example: For $h(x) = \frac{1}{x-2}$, the domain is all real numbers except $x=2$.

Exponential and Logarithmic Functions

Exponential Growth and Decay

Exponential functions model situations where quantities grow or decay at a constant percentage rate per unit time.

• General Form: $y = ab^{x}$, where $a$ is the initial value and $b$ is the growth ($b>1$) or decay ($0

• Applications: Population growth, radioactive decay, compound interest.

• Example: If a population doubles every 5 years, $P(t) = P_0 \cdot 2^{t/5}$.

Systems of Equations

Solving Systems Graphically and Algebraically

Systems of equations can be solved by graphing, substitution, or elimination.

• Graphical Solution: The solution is the point(s) where the graphs intersect.

• Algebraic Solution: Use substitution or elimination to find the values of variables that satisfy all equations.

• Example: Solve $\begin{cases} 2x + y = 4 \\ x - y = 1 \end{cases}$ by substitution or elimination.

Sequences and Series

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers defined by a rule. Arithmetic sequences have a constant difference; geometric sequences have a constant ratio.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Example: For $a_1 = 3$, $d = 2$, the sequence is $3, 5, 7, 9, ...$

Graph Interpretation and Analysis

Reading and Analyzing Graphs

Understanding graphs involves identifying key features such as intercepts, slopes, and points of intersection.

• Intercepts: Points where the graph crosses the axes.

• Slope: Rate of change of a linear function, $m = \frac{\Delta y}{\Delta x}$.

• Intersection Points: Solutions to systems of equations represented graphically.

• Example: Given a graph of two lines, the intersection point gives the solution to the system.

Additional Info

• Some questions require explaining the meaning of function values in context (e.g., interpreting $f(2)$ in a real-world scenario).

• Tables may be used to represent function values for specific inputs.

• Graphing and interpreting rational functions often involves identifying asymptotes and intercepts.