Pre-Calculus Strategy Guide: Essential Concepts and Graphing Techniques

Linear Functions

Definition and Properties

Linear functions are foundational in calculus and represent relationships with constant rates of change. The general form is $y = mx + b$, where m is the slope and b is the y-intercept.

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ for points $(x_1, y_1)$ and $(x_2, y_2)$.

• Slope-intercept form: $y = mx + b$.

• Point-slope form: $y - y_1 = m(x - x_1)$.

• Domain and Range: Both are $\mathbb{R}$ (all real numbers).

Strategy to Find Equations of Linear Functions

• If given a slope and a point, use point-slope form and rearrange.

• If given two points, calculate the slope, then use point-slope form.

Strategy for Graphing Linear Equations

• Start with the y-intercept $(0, b)$.

• Use the slope $m$ to find another point: move right 1 unit, up/down $m$ units.

• Draw a straight line through the points.

Example:

Graph $y = 2x + 1$: Start at $(0,1)$, move right 1, up 2 to $(1,3)$, draw the line.

Quadratics

General Form and Properties

Quadratic functions model parabolic relationships and are written as $ax^2 + bx + c$.

• Vertex/Standard form: $y = a(x-h)^2 + k$; vertex at $(h, k)$.

• Axis of symmetry: $x = h$.

• Domain: $\mathbb{R}$.

• Range: $y \geq k$ if $a > 0$, $y \leq k$ if $a < 0$.

Strategy for Graphing Quadratics

• Find the vertex $(h, k)$.

• Plot the vertex and axis of symmetry.

• Reflect points across the axis of symmetry.

• Draw the parabola through these points.

Strategy for Finding Roots of Quadratics

• Factoring: Fast if possible.

• Vertex form: Set $y=0$ and solve.

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

Example:

Find roots of $x^2 - 5x + 6 = 0$ by factoring: $(x-2)(x-3)=0$ so $x=2,3$.

Polynomials

Definition and End Behavior

Polynomials are sums of powers of $x$ with real coefficients. The degree and leading coefficient determine end behavior.

• General form: $p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$.

• If degree $n$ is even and $a_n > 0$, $p(x) \to \infty$ as $x \to \pm\infty$.

• If degree $n$ is odd and $a_n > 0$, $p(x) \to \infty$ as $x \to \infty$, $p(x) \to -\infty$ as $x \to -\infty$.

Strategy for Graphing General Shape of Factored Polynomial

• Plot roots and y-intercept.

• Determine root multiplicity (even: touch, odd: cross).

• Sketch end behavior.

Strategy to Find a Least Degree Polynomial Equation from a Graph

• Identify roots and their multiplicities.

• General form: $p(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}...$

• Solve for $a$ using a known point not at a root.

Rational Functions

Definition and Properties

Rational functions are quotients of polynomials: $f(x) = \frac{p(x)}{q(x)}$.

• Only integer powers of $x$ allowed.

• Domain: All $x$ except roots of $q(x)$.

• Vertical asymptote: $x =$ root of $q(x)$.

• Horizontal asymptote: determined by degrees of $p(x)$ and $q(x)$.

Rational Functions of the Form $f(x) = \frac{a}{x-b} + c$

• Vertical asymptote: $x = b$.

• Horizontal asymptote: $y = c$.

• Domain: $x \neq b$.

• Range: $y \neq c$.

Strategy for Graphing Rational Functions

• Draw asymptotes.

• Construct a table of values near asymptotes.

• Plot points and sketch the curve.

Strategy for Finding Equations of Rational Functions from Graphs

• Identify vertical and horizontal asymptotes.

• Determine form: $\frac{a}{x-b} + c$ or $\frac{a}{(x-b)^2} + c$.

• Solve for $a$ using a known point.

Root Functions

Definition and Properties

Root functions involve square roots or other radicals, typically $f(x) = a\sqrt{x-b} + c$.

• Graph starts at $(b, c)$.

• Domain: $x \geq b$.

• Range: $y \geq c$ if $a > 0$.

Strategy for Graphing a Root Function

• Plot starting point $(b, c)$.

• Construct a table of values for $x \geq b$.

• Plot points and connect smoothly.

Strategy for Finding an Equation of a Root Function from a Graph

• Find starting point $(b, c)$.

• Substitute into $f(x) = a\sqrt{x-b} + c$.

• Solve for $a$ using another point.

General Transformations

Definition and Types

Transformations shift, stretch, compress, or reflect graphs. The general form is $af(x-b) + c$.

• Vertical stretch/compression: $a$ scales $f(x)$ vertically.

• Horizontal shift: $b$ shifts $f(x)$ horizontally.

• Vertical shift: $c$ shifts $f(x)$ vertically.

• Reflection: Negative $a$ or $b$ reflects the graph.

Strategy for Graphing Transformations

• Apply stretches/compressions by scaling coordinates.

• Apply shifts by adding/subtracting from $x$ or $y$.

• Reflect as needed.

Example:

Graph $2f(x-3)+1$: stretch vertically by 2, shift right by 3, up by 1.

Absolute Values

Definition and Properties

The absolute value function $|x|$ returns the non-negative value of $x$.

• $|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

• Domain: $\mathbb{R}$.

• Range: $y \geq 0$.

Strategy for Graphing $|f(x)|$

• Graph $f(x)$.

• Reflect any part of the graph below the $x$-axis above it.

Example:

Graph $|x-2|$: V-shape with vertex at $(2,0)$.

Function Type	General Form	Key Features	Graph Shape
Linear	$y = mx + b$	Slope, y-intercept	Straight line
Quadratic	$y = ax^2 + bx + c$	Vertex, axis of symmetry	Parabola
Polynomial	$a_nx^n + ... + a_0$	Degree, end behavior	Varied curves
Rational	$\frac{p(x)}{q(x)}$	Asymptotes, domain	Hyperbola-like
Root	$a\sqrt{x-b} + c$	Starting point, domain	Curve from a point
Absolute Value	$|x|$	Vertex, reflection	V-shape

Additional info: These notes are suitable for students preparing for calculus, covering essential algebraic and graphing skills required for success in college-level calculus courses.