College Algebra Midterm Study Guide: Key Concepts and Practice

Equations and Inequalities

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, typically written in the form $ax^2 + bx + c = 0$. The solutions (roots) can be found using factoring, completing the square, or the quadratic formula.

• General Form: $ax^2 + bx + c = 0$

• Given Solutions: If the solutions are $r_1$ and $r_2$, the equation can be written as $a(x - r_1)(x - r_2) = 0$.

• Example: For solutions (2, -1), the equation is $(x - 2)(x + 1) = 0$ or $x^2 - x - 2 = 0$.

Linear Inequalities

Linear inequalities involve expressions with a variable to the first power. The solution is often represented in interval notation.

• Solving: Isolate the variable using algebraic operations, remembering to reverse the inequality sign when multiplying or dividing by a negative number.

• Interval Notation: Expresses the set of solutions, e.g., $(-\infty, -1/2]$.

• Example: Solve $(x + 2)/(x - 5) \leq 3x + 2$.

Functions and Their Graphs

Definition of a Function

A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: The set $\{(1, 11), (1, 22), (2, 8), (3, 10)\}$ is not a function because the input 1 maps to two different outputs (11 and 22).

Tables and Functions

Tables can be used to determine if a relation is a function by checking if each input value corresponds to only one output value.

x	y
-5	9
-4	17
-9	17
-17	17

• Domain: $\{-5, -4, -9, -17\}$

• Range: $\{9, 17\}$

• Function? Yes, if each x-value is unique.

Linear and Quadratic Functions

Equations of Lines

The equation of a line can be written in slope-intercept form ($y = mx + b$) or point-slope form ($y - y_1 = m(x - x_1)$).

• Slope (m): The rate of change of the line.

• Example: For slope $m = 2$ and passing through $(-1, 1/2)$, use point-slope form first, then solve for $y$.

Equations of Circles

The standard form for the equation of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

• Example: Center $(-8, -16)$, $r = \sqrt{8}$: $(x + 8)^2 + (y + 16)^2 = 8$

Transformations of Functions

Square Root and Absolute Value Functions

Transformations include translations, reflections, stretches, and compressions.

• Vertical Stretch: $h(x) = 3\sqrt{x + 4}$ is a vertical stretch by 3 and a horizontal shift left by 4 units of $f(x) = \sqrt{x}$.

• Absolute Value: $g(x) = |5x - 5| - 2/2$ is a transformation of $f(x) = |5x|$.

Function Operations and Composition

Function Composition

Given two functions $f$ and $g$, the composition $f(g(x))$ means to substitute $g(x)$ into $f(x)$.

• Example: If $f(x) = 71x^2 - 2$, $g(x) = x + 10$, then $f(g(x)) = 71(x + 10)^2 - 2$.

Quadratic Functions

Vertex Form

The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

• Example: For vertex $(-1, -1)$ and same shape as $f(x) = 13x^2$, $y = 13(x + 1)^2 - 1$.

Polynomial and Rational Functions

Graphing Polynomial Functions

Polynomial functions can be graphed by finding their zeros, end behavior, and plotting key points.

• Example: $f(x) = (x - 17)(x - 29)$ is a quadratic with zeros at $x = 17$ and $x = 29$.

Dividing Polynomials

Polynomial long division is used to divide one polynomial by another, resulting in a quotient and a remainder.

• Example: Divide $4x^2 + 2x^2 + 8x^3$ by $5x^2 + 3$.

Zeros of Polynomial Functions

To find all zeros of a polynomial, use factoring, the Rational Root Theorem, or the quadratic formula.

• Example: If $x = 7$ is a zero of $f(x) = x^3 - 13x^2 - 14$, factor to find other zeros.

End Behavior of Polynomials

The end behavior of a polynomial function is determined by the degree and leading coefficient.

• Leading Coefficient Test: For $f(x) = 2x^4 + 19x^2 - 3x - 2$, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to \infty$ (since degree is even and leading coefficient is positive).

Rational Functions

Graphing Rational Functions

Rational functions are of the form $f(x) = \frac{P(x)}{Q(x)}$. Key features include vertical and horizontal asymptotes, holes, and intercepts.

• Vertical Asymptotes: Values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

• Example: $f(x) = \frac{(x^2 - 8x - 16)}{(4x^2 - 5x)}$

Asymptotes and Intercepts

• Vertical Asymptote: Set denominator equal to zero and solve for $x$.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• x-intercept: Set numerator equal to zero and solve for $x$.

Summary Table: Key Features of Functions

Function Type	Standard Form	Key Features
Linear	$y = mx + b$	Slope, y-intercept
Quadratic	$y = ax^2 + bx + c$	Vertex, axis of symmetry, zeros
Circle	$(x - h)^2 + (y - k)^2 = r^2$	Center, radius
Rational	$f(x) = \frac{P(x)}{Q(x)}$	Asymptotes, intercepts, holes

Additional info: This guide covers core College Algebra topics including equations, inequalities, functions, graphing, transformations, and polynomial/rational function analysis, as reflected in the midterm questions.