Graphs, Functions, and Models

Distance and Midpoint Formulas

The distance formula and midpoint formula are essential tools for analyzing points in the coordinate plane.

• Distance Formula: Calculates the distance between two points and .

• Midpoint Formula: Finds the midpoint between two points and .

• Example: For points (2, 3) and (6, 7):

  • Distance:

  • Midpoint:

Solving Equations Algebraically and Graphically

Equations can be solved using algebraic manipulation or by interpreting their graphs.

• Linear Equations: Equations of the form ; solution is .

• Quadratic Equations: Equations of the form ; solutions found using factoring, completing the square, or the quadratic formula:

• Exponential Equations: Equations involving terms like ; often solved using logarithms.

• Logarithmic Equations: Equations involving ; solved by exponentiating both sides.

• Graphical Solutions: The solution to is the x-intercept of the graph of .

• Example: Solve algebraically: ; graphically, find where the line crosses .

Solving Inequalities

Inequalities express a range of possible solutions and can be solved algebraically or graphically.

• Linear Inequalities: or ; solve like equations, then graph solution on a number line.

• Quadratic Inequalities: ; solve by finding roots and testing intervals.

• Example: Solve ; roots at ; solution is .

More on Functions

Identifying and Graphing Functions

Functions can be linear, quadratic, exponential, or logarithmic. Their graphs have distinct shapes and properties.

• Linear Functions: ; graph is a straight line.

• Quadratic Functions: ; graph is a parabola.

• Exponential Functions: ; graph increases or decreases rapidly.

• Logarithmic Functions: ; graph increases slowly, undefined for .

• Example: Graph ; passes through (0,1), (1,2), (2,4).

Domain and Range of Functions

The domain is the set of all possible input values; the range is the set of all possible output values.

• From Graph: Identify the x-values (domain) and y-values (range) covered by the graph.

• From Equation: Analyze restrictions (e.g., denominator cannot be zero, argument of square root must be non-negative).

• Example: ; domain: , range: .

x- and y-Intercepts of a Function

Intercepts are points where the graph crosses the axes.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

• Example: For , x-intercepts at ; y-intercept at .

Even, Odd, and Neither Functions

Functions can be classified based on their symmetry.

• Even Function: ; symmetric about the y-axis.

• Odd Function: ; symmetric about the origin.

• Neither: Does not satisfy either property.

• Example: is even; is odd.

Transformations of Graphs and Equations

Transformations alter the position and shape of a graph.

• Vertical Shift: shifts up/down by units.

• Horizontal Shift: shifts right by units.

• Vertical Compression/Stretch: compresses if , stretches if .

• Horizontal Compression/Stretch: compresses if , stretches if .

• Example: is a parabola shifted right 2 units and up 3 units.

Quadratic, Exponential, and Logarithmic Functions

Linear Models

Linear models describe relationships with constant rates of change.

• Equation:

• Application: Used in economics, physics, and statistics to model trends.

• Example: Predicting cost based on quantity:

Composition of Functions

Composition combines two functions into one by substituting one into the other.

• Notation:

• Example: If , , then

Inverse Functions

An inverse function reverses the effect of the original function.

• Definition: for all in the domain of .

• Finding the Inverse: Swap and in , then solve for .

• Example: ; inverse:

Equations for Inverse of One-to-One Functions

One-to-one functions have unique inverses.

• Process: Set , solve for in terms of , then write .

• Example: ; ;

Solving Logarithmic Expressions

Logarithmic expressions can be simplified using properties of logarithms.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Example: because

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Methods: Substitution, elimination, and matrix methods.

• Example: Solve:

  Solution: Add equations: ; then .

Additional info: Some context and examples have been inferred to ensure completeness and clarity for exam preparation.