Functions and Their Graphs

Graphing Radical and Piecewise Functions

Understanding how to graph functions, including those involving radicals and piecewise definitions, is a key skill in precalculus. This involves plotting points, identifying intercepts, and analyzing transformations.

• Radical Functions: Functions involving roots, such as , often have restricted domains and unique shapes.

• Piecewise Functions: Defined by different expressions over different intervals, requiring careful attention to endpoints and continuity.

• Intercepts: x-intercepts are points where ; y-intercepts are points where .

Example: For , select values of , compute , and plot the points. The graph is a reflection and vertical shift of the parent square root function.

Circles in the Coordinate Plane

Equation of a Circle Given a Diameter

The equation of a circle with center and radius is:

• To find the center, use the midpoint formula between the endpoints of the diameter: .

• The radius is half the distance between the endpoints.

Example: For endpoints and , the center is and the radius is $2$.

Linear Equations and Slopes

Writing Equations of Lines

Lines can be described using the point-slope or slope-intercept forms:

• Slope-Intercept Form:

• Point-Slope Form:

• Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

Example: A line with slope through : .

Function Operations and Domains

Combining Functions and Finding Domains

Functions can be added, subtracted, multiplied, divided, and composed. The domain of a function is the set of all input values for which the function is defined.

• Sum:

• Difference:

• Product:

• Quotient: ,

• Composition:

Example: If and , the domain excludes values making any denominator zero.

Analyzing Functions: Extrema, Intervals, and One-to-One

Graphical Analysis and Transformations

Analyzing a function's graph involves identifying:

• Relative Extrema: Local maximum and minimum points.

• Intervals of Increase/Decrease: Where the function rises or falls as increases.

• Constant Intervals: Where the function remains unchanged.

• One-to-One Functions: Each -value corresponds to exactly one -value. Passes the horizontal line test.

• Transformations: Shifts, stretches, compressions, and reflections applied to parent functions.

Example: For , analyze each piece separately for intervals and extrema.

Average Rate of Change

Definition and Calculation

The average rate of change of a function from to is:

This measures the "slope" of the secant line between two points on the graph.

Verifying One-to-One Functions Algebraically

Algebraic Test for One-to-One

A function is one-to-one if implies . To verify, set and solve for and .

Example: For , if then so .

Summary Table: Function Types and Properties