MAT151 Precalculus: Core Concepts and Procedures for College Algebra

Module I: Function Fundamentals (The Language of Algebra)

Core Conceptual Definitions

• Function as a "Coffee Machine": A function is a rule that assigns each input exactly one output. For every value of x, there is only one corresponding y value. This is analogous to a coffee machine where each button (input) produces only one type of coffee (output).

• Function vs. Relation: A relation is any set of ordered pairs. A function is a special type of relation where no input is paired with more than one output.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point. For example, a circle fails this test, so it is not a function.

Step-by-Step Procedures

• Solving Linear Equations by Graphing:

  1. Rewrite the equation as two separate equations: e.g., for $2x - 4 = 6$, write $y = 2x - 4$ and $y = 6$.

  2. Graph both equations. $y = 2x - 4$ is a line with slope 2 and y-intercept -4; $y = 6$ is a horizontal line.

  3. The intersection point gives the solution. Here, the lines intersect at $(5, 6)$, so $x = 5$ is the solution.

• Writing Domain and Range in Interval Notation:

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

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

  • Interval Notation: Use parentheses $( )$ for values not included (open intervals), square brackets $[ ]$ for values included (closed intervals), and the union symbol $\cup$ to combine intervals. Example: $(-\infty, -3/4] \cup [7/4, \infty)$.

Comparison Table: Toolkit Functions Part 1

Function Name	Equation	Basic Shape	Defining Characteristics
Identity	$f(x) = x$	Diagonal Line	Passes through $(0,0)$; output equals input.
Absolute Value	$f(x) = |x|$	V-shaped	Always non-negative; vertex at $(0,0)$.

Common Pitfalls & Self-Correction

• Check for repeated $x$-values with different $y$-values in data sets; this indicates a relation, not a function.

Module II: Function Analysis & Long-Run Behavior

Core Conceptual Definitions

• Average Rate of Change (ARC): The slope of the line connecting two points on a function, calculated as $m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$.

• Increasing/Decreasing: A function is increasing where its graph rises as you move left to right, and decreasing where it falls.

• Local Extrema: Local maxima are peaks, and local minima are valleys where the function changes direction.

Step-by-Step Procedures

• Determining End Behavior of Polynomials:

  1. Focus on the leading term $ax^n$.

  2. If $n$ is even, both ends of the graph point in the same direction; if odd, they point in opposite directions.

  3. If $a > 0$, the right end rises ($f(x) \to \infty$); if $a < 0$, the right end falls ($f(x) \to -\infty$).

• Finding Intercepts:

  • y-intercept: Set $x = 0$ and solve for $y$.

  • x-intercepts: Set $y = 0$ and solve for $x$ (these are the roots).

Comparison Table: Polynomial Terminology

Example	Name	Leading Coefficient	Constant Term	Degree
$x + 9$	Binomial	1	9	1
$3x^5$	Monomial	3	0	5
$4x^8 - 6x^5 + 3x^2 + 7$	Trinomial	4	7	8

Common Pitfalls & Self-Correction

• Use fractions for non-terminating decimals (e.g., $1/3$ instead of $0.333...$) to avoid rounding errors in interval notation.

Module III: Operations & Polynomial Division

Core Conceptual Definitions

• Function Composition: The process of applying one function to the results of another, written as $(f \circ g)(x) = f(g(x))$.

• Decomposition: Breaking a complex function into simpler inner and outer functions.

Step-by-Step Procedures

• Polynomial Long Division:

  1. Divide the leading term of the dividend by the leading term of the divisor.

  2. Multiply the entire divisor by this result and subtract from the dividend.

  3. Repeat until the degree of the remainder is less than the divisor.

  4. Placeholder Rule: Insert zero-coefficient terms for missing degrees to keep columns aligned (e.g., $x^4 + 0x^3 - 2x^2 + 0x + 3$).

• Synthetic Division:

  1. For division by $(x - c)$, use $c$ in the synthetic division setup.

  2. List coefficients, drop the first, multiply by $c$, add to the next, and repeat.

Comparison Table: Factor vs. Zero

Concept	Example	Meaning	Relationship
Factor	$(x + 4)$	Algebraic expression	If division by $(x + 4)$ yields remainder 0, it is a factor.
Zero	$-4$	x-intercept	If $-4$ is a zero, then $(x + 4)$ is a factor.

Common Pitfalls & Self-Correction

• Always subtract the entire expression in long division; change the sign of every term before adding.

Module IV: Function Families (Quadratic & Power Functions)

Core Conceptual Definitions

• Power Functions: Functions of the form $f(x) = kx^p$, where $k$ and $p$ are constants. These are the building blocks of polynomials.

• Quadratic Functions: Standard form is $f(x) = ax^2 + bx + c$; vertex form is $f(x) = a(x - h)^2 + k$.

• "Skinny vs. Wide" Modifier: The value of $a$ in the vertex form determines the vertical stretch or compression. If $|a| > 1$, the parabola is "skinny"; if $0 < |a| < 1$, it is "wide".

Step-by-Step Procedures

• Converting General Form to Vertex Form:

  1. Given $f(x) = 3x^2 + 6x + 8$, find $h$ using $x = -b/(2a)$. Here, $h = -6/(2 \times 3) = -1$.

  2. Find $k$ by evaluating $f(h)$: $f(-1) = 3(-1)^2 + 6(-1) + 8 = 5$.

  3. Write vertex form: $f(x) = 3(x + 1)^2 + 5$.

• Application Example (Rocket Launch):

  • Given $h(t) = -4.9t^2 + 310t + 332$.

  • Maximum height at $t = -310/(2 \times -4.9) \approx 31.63$ s; $h(31.63) \approx 5235.1$ m.

  • Splashdown: Set $h(t) = 0$ and solve using the quadratic formula; $t \approx 64.3$ s.

Comparison Table: End Behavior Matrix for $f(x) = kx^p$

Exponent ($p$)	Coefficient ($k$)	Left End ($x \to -\infty$)	Right End ($x \to \infty$)
Even	Positive	$f(x) \to \infty$	$f(x) \to \infty$
Even	Negative	$f(x) \to -\infty$	$f(x) \to -\infty$
Odd	Positive	$f(x) \to -\infty$	$f(x) \to \infty$
Odd	Negative	$f(x) \to \infty$	$f(x) \to -\infty$

Common Pitfalls & Self-Correction

• Apply stretches, compressions, and reflections before horizontal or vertical shifts to maintain correct vertex position.

Module V: Rational Functions & Systems

Core Conceptual Definitions

• Asymptotes: Lines that the graph approaches but never crosses. Vertical asymptotes occur where the denominator is zero and not canceled by the numerator; horizontal asymptotes describe end behavior.

• Multiplicity Diagnostic: If a denominator factor's exponent is odd, the graph crosses the asymptote; if even, it stays on one side.

Step-by-Step Procedures

• Identifying Holes vs. Vertical Asymptotes:

  1. Factor numerator and denominator completely.

  2. If a factor cancels (e.g., $(x - 4)$ in both), there is a hole at $x = 4$.

  3. If a factor remains in the denominator only (e.g., $(x + 2)$), there is a vertical asymptote at $x = -2$.

• Solving Rational Inequalities:

  1. Find critical values where numerator or denominator equals zero.

  2. Test intervals between these values to determine where the inequality holds.

• Average Cost Application:

  • Given $A(x) = \frac{6x + 125}{x}$, to find $x$ for $A(x) = 11$, solve $11x = 6x + 125 \implies 5x = 125 \implies x = 25$ units.

  • Horizontal asymptote at $y = 6$; as $x \to \infty$, $A(x) \to 6$.

Comparison Table: Horizontal Asymptote Shortcut Table

Degree Comparison	Resulting Asymptote	End Behavior ($x \to \infty$)
Num. Degree < Denom. Degree	$y = 0$	$f(x) \to 0$
Num. Degree = Denom. Degree	$y = a/b$	$f(x) \to$ Ratio of coefficients
Num. Degree > Denom. Degree	None	$f(x) \to \infty$ or $-\infty$

Common Pitfalls & Self-Correction

• Horizontal shifts in rational functions are counter-intuitive: $g(x) = \frac{1}{x - 3} + 4$ shifts right by 3 and up by 4. Always check asymptote positions before sketching.