Chapter F: Foundations – A Prelude to Functions

F.1 The Distance and Midpoint Formulas

This section introduces the distance and midpoint formulas, essential tools for analytic geometry and understanding the coordinate plane.

• Distance Formula: The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the plane is given by: Key Point: This formula is derived from the Pythagorean Theorem.

• Midpoint Formula: The midpoint \(M\) of the segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is: Key Point: The midpoint is the average of the x-coordinates and y-coordinates.

• Example: Find the distance and midpoint between \((1, 2)\) and \((4, 6)\):

  • Distance:

  • Midpoint:

F.2 Intercepts; Symmetry; Graphing Key Equations

This section covers how to find intercepts and test for symmetry in equations and graphs.

• Intercepts:

  • x-intercept: Set \(y = 0\) and solve for \(x\).

  • y-intercept: Set \(x = 0\) and solve for \(y\).

• Symmetry Tests:

  • y-axis symmetry: Replace \(x\) with \(-x\); if the equation is unchanged, the graph is symmetric about the y-axis.

  • x-axis symmetry: Replace \(y\) with \(-y\); if unchanged, symmetric about the x-axis.

  • Origin symmetry: Replace both \(x\) and \(y\) with \(-x\) and \(-y\); if unchanged, symmetric about the origin.

• Example: For \(y = x^2\):

  • y-intercept: \(x = 0 \Right arrow y = 0\)

  • x-intercept: \(y = 0 \Right arrow x = 0\)

  • Symmetry: \(y = (-x)^2 = x^2\) (y-axis symmetry)

F.3 Lines

This section focuses on the properties and equations of lines, including slope, forms of equations, and relationships between lines.

• Slope of a Line: The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• Forms of Linear Equations:

  • Point-Slope Form:

  • Slope-Intercept Form:

  • General Form:

  • Vertical Line:

  • Horizontal Line:

• Parallel and Perpendicular Lines:

  • Parallel: Same slope, different y-intercept.

  • Perpendicular: Product of slopes is \(-1\):

• Example: Find the equation of the line through \((2, 3)\) with slope 4:

  • Point-slope:

  • Slope-intercept:

Chapter 1: Functions and Their Graphs

1.1 Functions

This section introduces the concept of functions, their evaluation, and domain determination.

• Definition: A function is a relation in which each input (x-value) has exactly one output (y-value).

• Evaluating Functions: Substitute the given value into the function.

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

  • Square roots: The expression under the root must be non-negative.

  • Rational functions: The denominator cannot be zero.

• Operations on Functions: Sum, difference, product, quotient, and difference quotient.

  • Difference Quotient:

• Example: If , find and the domain.

  • Domain: All real numbers

1.2 The Graph of a Function

This section explains how to interpret and analyze the graph of a function.

• Vertical-Line Test: A graph represents a function if no vertical line intersects it more than once.

• Reading Graphs: Use the graph to find values, intercepts, domain, range, and intervals where the function is positive or negative.

• Example: For , the graph passes the vertical-line test and is always non-negative.

1.3 Properties of Functions

This section covers even and odd functions, as well as intervals of increase, decrease, and constancy.

• Even Functions: for all in the domain. Graph is symmetric about the y-axis.

• Odd Functions: for all in the domain. Graph is symmetric about the origin.

• Increasing/Decreasing/Constant: Use the graph to determine intervals where the function rises, falls, or remains flat.

• Maxima and Minima: Highest (maximum) or lowest (minimum) points on the graph; can be local or absolute.

• Example: is odd; is even.

1.4 Library of Functions; Piecewise-defined Functions

This section introduces basic functions and piecewise-defined functions.

• Library of Functions: Includes linear, quadratic, cubic, square root, absolute value, reciprocal, etc.

• Piecewise-defined Functions: Functions defined by different expressions over different intervals.

  • Example:

• Graphing: Graph each piece over its specified interval.

1.5 Graphing Techniques: Transformations

This section discusses how to graph functions using transformations and how to write equations for transformed graphs.

• Transformations:

  • Vertical shift: shifts up/down

  • Horizontal shift: shifts right/left

  • Reflection: reflects over x-axis; reflects over y-axis

  • Stretch/Compression: stretches (|a| > 1) or compresses (0 < |a| < 1) vertically

• Finding Points: Apply transformations to known points on the parent function.

• Example: is shifted right 3, stretched by 2, and up 1.

1.6 Building Mathematical Models from Verbal Descriptions

This section focuses on translating real-world scenarios into mathematical models using functions and geometric formulas.

• Geometric Formulas: Know basic formulas such as:

  • Area of a rectangle:

  • Circumference of a circle:

  • Area of a circle:

• Modeling: Express relationships described in words as functions or equations.

• Example: "The area of a rectangle is a function of its length if the width is fixed."

1.7 Building Mathematical Models Using Variation

This section introduces direct and inverse variation and their applications.

• Direct Variation: , where is the constant of proportionality.

• Inverse Variation: , where is the constant of proportionality.

• Applications: Use these models to solve real-world problems involving proportional relationships.

• Example: If varies directly with and when , then and .

Additional info: This guide covers all foundational topics and function concepts likely to appear on Exam 1, including key formulas, properties, and modeling strategies. Students should be comfortable with both algebraic manipulation and interpreting graphs.