College Algebra: Functions, Quadratics, and Polynomials Study Guide

Functions and Their Operations

Definition and Evaluation of Functions

In algebra, a function is a rule that assigns each input exactly one output. Functions can be combined and manipulated in various ways.

• Function notation: If $f(x)$ and $g(x)$ are functions, then $(f+g)(x)$, $(f-g)(x)$, and $(f imes g)(x)$ represent their sum, difference, and product, respectively.

• Example: If $r(x) = x + 3$ and $s(x) = 2x^2$, then:

  • $(s - r)(x) = 2x^2 - (x + 3) = 2x^2 - x - 3$

  • $(sr)(x) = (x + 3)(2x^2) = 2x^3 + 6x^2$

  • $(s + r)(-3) = 2(-3)^2 + (-3) + 3 = 18$

Composition of Functions

The composition of functions $f$ and $g$ is written as $(f igcirc g)(x) = f(g(x))$.

• Example: If $q(x) = 3x - 1$ and $r(x) = -4x^2 + 1$, then:

  • $(r igcirc q)(-5) = r(q(-5)) = r(-16) = -4(-16)^2 + 1 = 65$

  • $(q igcirc r)(-5) = q(r(-5)) = q(-99) = 3(-99) - 1 = -298$

Quadratic Functions and Applications

Quadratic Equations and Their Graphs

A quadratic function has the form $f(x) = ax^2 + bx + c$, where $a eq 0$. Its graph is a parabola.

• Vertex: The vertex of $f(x) = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$.

• Axis of symmetry: The line $x = -\frac{b}{2a}$ divides the parabola into two symmetric parts.

• Intercepts:

  • x-intercepts: Solve $ax^2 + bx + c = 0$.

  • y-intercept: Set $x = 0$; $f(0) = c$.

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• Example: For $y = (x + 3)(x + 1)$:

  • Vertex: $(-2, 1)$

  • Axis of symmetry: $x = -2$

  • x-intercepts: $x = -3, -1$

  • y-intercept: $y = -3$

Quadratic Applications

Quadratic functions model real-world scenarios, such as projectile motion.

• Example: The height of a ball thrown from a building: $h = 67 - 5t - 5t^2$

  • To find when the ball hits the ground, solve $h = 0$ for $t$.

  • Solution: $t = 3.19$ seconds (rounded to the nearest hundredth).

Polynomial Functions

Definition and Properties

A polynomial function is an expression of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$.

• Degree: The highest power of $x$ in the polynomial.

• Leading coefficient: The coefficient of the term with the highest degree.

• Example: For $w^8 - 6 + 3w + 5w^6$:

  • Leading coefficient: $1$

  • Degree: $8$

Zeros and Multiplicity

The zeros of a polynomial are the values of $x$ for which $f(x) = 0$. The multiplicity of a zero is the number of times it appears as a root.

• Example: For $f(x) = 4x(x-7)(x+13)^2(x+5)^3$:

  • Zero(s) of multiplicity one: $0, 7$

  • Zero(s) of multiplicity two: $-13$

  • Zero(s) of multiplicity three: $-5$

End Behavior of Polynomials

The end behavior of a polynomial describes how the function behaves as $x$ approaches infinity or negative infinity.

• If the degree is even and the leading coefficient is positive, both ends rise.

• If the degree is even and the leading coefficient is negative, both ends fall.

• If the degree is odd and the leading coefficient is positive, left falls and right rises.

• If the degree is odd and the leading coefficient is negative, left rises and right falls.

Graphing Polynomial Functions

To graph a polynomial:

1. Identify zeros and their multiplicities.

2. Determine end behavior.

3. Find y-intercept ($f(0)$).

4. Plot points and connect them smoothly, respecting the behavior at each zero.

Polynomial Division

Polynomials can be divided using long division or synthetic division. The result is a quotient and a remainder.

• Example: Divide $f(x) = (x^2 + 4x + 5)$ by $(x + 2)$:

  • Quotient: $x + 2$

  • Remainder: $1$

Solving Polynomial Inequalities

Graphical Solution

To solve $x^2 ext{≤} 6x$:

• Rewrite as $x^2 - 6x ext{≤} 0$.

• Find zeros: $x(x - 6) = 0 ightarrow x = 0, 6$.

• Test intervals to determine where the inequality holds.

• Graph the solution on a number line.

Analyzing Graphs of Polynomial Functions

Increasing/Decreasing Intervals and Local Maxima

From a graph, you can determine where a function is increasing, decreasing, and where it has local maxima or minima.

• Increasing intervals: Where the graph rises as $x$ increases.

• Local maxima: Points where the function reaches a peak locally.

• Example: For a given graph, the function is increasing on $(-6, 3)$, $(7, 9)$, and has local maxima at $x = 6, 3, 9$.

Sign of Leading Coefficient and Degree

• Sign of leading coefficient: Determines the end behavior.

• Degree: The number of turning points is at most one less than the degree.

Summary Table: Key Properties of Polynomial Functions

Property	Description
Degree	Highest exponent of $x$
Leading Coefficient	Coefficient of highest degree term
Zeros	Values of $x$ where $f(x) = 0$
Multiplicity	Number of times a zero is repeated
End Behavior	Behavior as $x \to \pm\infty$
Vertex (Quadratic)	Maximum or minimum point
Axis of Symmetry (Quadratic)	Vertical line through vertex

Additional info:

• All examples and explanations are based on standard College Algebra curriculum.

• Some values and steps were inferred for completeness and clarity.