Intercepts, Domain & Range, Asymptotes, and Behavior of Functions: College Algebra Study Notes

Solving Higher-Order Polynomial Equations

Finding Solutions to Polynomial Equations

Solving higher-order polynomial equations involves finding all values of x that satisfy the equation. These values are called roots or zeros of the polynomial.

• Example: Solve $3x^2(x^2-16) = 0$

• Set each factor equal to zero: $3x^2 = 0$ and $x^2-16 = 0$

• Solutions: $x = 0$, $x = 4$, $x = -4$

• Solution set: $\{ -4, 0, 4 \}$

Intercepts of a Function

Definition of Intercepts

• y-intercept: The y-coordinate of the point where the graph crosses or touches the y-axis. Found by evaluating $f(0)$.

• x-intercept: The x-coordinate(s) of the point(s) where the graph crosses or touches the x-axis. Found by solving $f(x) = 0$.

Finding Intercepts

• A function can have only one y-intercept, which exists if $x = 0$ is in the domain of the function.

• A function can have multiple x-intercepts (real zeros), found by solving $f(x) = 0$.

Example: Find the intercepts of $f(x) = -3x + 2$

• y-intercept: $f(0) = -3(0) + 2 = 2$

• x-intercept: Set $-3x + 2 = 0 \implies x = \frac{2}{3}$

Domain and Range of Functions

Definitions

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) that the function can produce.

Example: Determining Domain and Range from a Graph

• Given a graph, the domain is the interval on the x-axis covered by the graph.

• The range is the interval on the y-axis covered by the graph.

• Example: If a graph extends from $x = a$ to $x = b$, then Domain is $[a, b]$.

• If the graph reaches from $y = c$ to $y = d$, then Range is $[c, d]$.

Asymptotes

Vertical Asymptotes

• A vertical asymptote is a vertical line $x = a$ where the function increases or decreases without bound as $x$ approaches $a$.

• Definition: $x = a$ is a vertical asymptote of $f(x)$ if $f(x) \to \infty$ or $f(x) \to -\infty$ as $x \to a$.

Horizontal Asymptotes

• A horizontal asymptote is a horizontal line $y = b$ that the graph of a function approaches as $x$ goes to $\infty$ or $-\infty$.

• Definition: $y = b$ is a horizontal asymptote of $f(x)$ if $f(x) \to b$ as $x \to \infty$ or $x \to -\infty$.

Behavior of Functions: Increasing, Decreasing, and Constant Intervals

Definitions

• Increasing: A function $f(x)$ is increasing on an interval if for any $x_1 < x_2$ in the interval, $f(x_1) < f(x_2)$.

• Decreasing: A function $f(x)$ is decreasing on an interval if for any $x_1 < x_2$ in the interval, $f(x_1) > f(x_2)$.

• Constant: A function $f(x)$ is constant on an interval if for any $x_1, x_2$ in the interval, $f(x_1) = f(x_2)$.

Example: Determining Intervals from a Graph

• Given a graph, identify where the function rises (increasing), falls (decreasing), or remains flat (constant).

• Example: If $f(x)$ increases from $x = -\infty$ to $x = -2$ and from $x = 3$ to $\infty$, then the function is increasing on $(-\infty, -2)$ and $(3, \infty)$.

Relative Maximum and Minimum

Definitions

• Relative Maximum: A point $(c, f(c))$ where $f(x)$ changes from increasing to decreasing. $f(c)$ is greater than all nearby values.

• Relative Minimum: A point $(c, f(c))$ where $f(x)$ changes from decreasing to increasing. $f(c)$ is less than all nearby values.

Example: Identifying Relative Extrema from a Graph

• Look for peaks (relative maxima) and valleys (relative minima) on the graph.

• Example: If $f(x)$ has a minimum at $x = -3$ and $x = 5$, then these are points where the function changes from decreasing to increasing.

Summary Table: Key Concepts

Concept	Definition	How to Find
y-intercept	Point where graph crosses y-axis	Evaluate $f(0)$
x-intercept	Point(s) where graph crosses x-axis	Solve $f(x) = 0$
Domain	All possible x-values	Analyze graph or function's formula
Range	All possible y-values	Analyze graph or function's formula
Vertical Asymptote	Vertical line $x = a$ where $f(x)$ diverges	Find values where denominator is zero (for rational functions)
Horizontal Asymptote	Horizontal line $y = b$ approached as $x \to \infty$	Analyze end behavior of $f(x)$
Increasing Interval	Interval where $f(x)$ rises	Check where $f(x_1) < f(x_2)$ for $x_1 < x_2$
Decreasing Interval	Interval where $f(x)$ falls	Check where $f(x_1) > f(x_2)$ for $x_1 < x_2$
Constant Interval	Interval where $f(x)$ is flat	Check where $f(x_1) = f(x_2)$
Relative Maximum	Peak point on graph	Where $f(x)$ changes from increasing to decreasing
Relative Minimum	Valley point on graph	Where $f(x)$ changes from decreasing to increasing

Additional info: These notes are based on standard College Algebra curriculum and expand on handwritten and graphical examples provided in the original materials.