College Algebra: Midterm Exam 3 Sample Questions Study Guide

Polynomial Functions

Roots and the Rational Root Theorem

Polynomial equations can be solved by finding their roots, which are the values of the variable that make the polynomial equal to zero. The Rational Root Theorem helps identify possible rational roots of a polynomial equation with integer coefficients.

• Rational Root Theorem: If $f(x) = a_nx^n + ... + a_0$ has integer coefficients, any rational root $\frac{p}{q}$ must have $p$ dividing $a_0$ and $q$ dividing $a_n$.

• Example: For $f(x) = x^3 + 8x^2 + 11x - 20$, possible rational roots are factors of $-20$ divided by factors of $1$.

• Finding roots: Substitute possible values into $f(x)$ to check which ones yield zero.

• Factoring: Once a root is found, use polynomial division or synthetic division to factor the polynomial further.

Additional info: Synthetic division is a shortcut for dividing polynomials by linear factors.

Turning Points and End Behavior

The graph of a polynomial function has certain characteristics:

• Turning Points: The maximum number of turning points is one less than the degree of the polynomial.

• End Behavior: Determined by the leading term. For $f(x) = ax^n$, if $a > 0$ and $n$ is even, both ends go up; if $a < 0$ and $n$ is even, both ends go down; if $n$ is odd, ends go in opposite directions.

Graphs of Equations

Graphing Polynomial and Rational Functions

Graphing involves plotting points, identifying intercepts, and analyzing asymptotes and behavior.

• Intercepts: Find where the graph crosses the axes by setting $x=0$ and $y=0$.

• Asymptotes: Rational functions may have vertical and horizontal asymptotes.

• Example: For $g(x) = \frac{8x}{x^2 - 1}$, vertical asymptotes at $x = 1$ and $x = -1$ (where denominator is zero), horizontal asymptote at $y = 0$ (degree of denominator higher than numerator).

Functions

Function Notation and Operations

Functions can be combined and evaluated using algebraic operations.

• Function notation: $f(x)$ represents the output of function $f$ for input $x$.

• Operations: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x)g(x)$, $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$.

• Domain: The set of all input values for which the function is defined.

• Example Table:

x	f(x)	g(x)
-2	3	4
-1	1	2
0	0	1
1	-1	0
2	-2	-1

• Example: $(f+g)(-2) = f(-2) + g(-2) = 3 + 4 = 7$

• Domain of $\frac{f}{g}$: All $x$ where $g(x) \neq 0$.

Inverse Functions

An inverse function reverses the effect of the original function. If $f(x)$ maps $x$ to $y$, then $f^{-1}(y)$ maps $y$ back to $x$.

• Finding inverses: Solve $y = f(x)$ for $x$ in terms of $y$.

• Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$.

Rational Functions

Properties and Graphs

Rational functions are quotients of polynomials. Their graphs can have asymptotes and discontinuities.

• Vertical asymptotes: Where the denominator is zero and the numerator is not zero.

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

• Example: For $g(x) = \frac{8x}{x^2 - 1}$, vertical asymptotes at $x = 1$ and $x = -1$.

Systems of Equations

Solving Linear Systems

Systems of equations can be solved graphically or algebraically (substitution, elimination).

• Graphical solution: The intersection point of two lines represents the solution.

• Example: Solve $2x + y = 4$ and $x - y = 1$ by graphing or substitution.

• Algebraic solution: Substitute or eliminate variables to find $x$ and $y$.

Equations & Inequalities

Solving Quadratic Equations

Quadratic equations are of the form $ax^2 + bx + c = 0$ and can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Express the quadratic as a product of two binomials.

• Completing the square: Rearrange and solve for $x$.

Additional info:

• Some questions ask for explanations of function properties, such as symmetry and domain.

• Graphical analysis is important for understanding solutions to systems and behavior of functions.