1. Functions & Graphs

Definition

A function is a relation where each input (x) has exactly one output (y).

• Vertical Line Test: If a vertical line touches the graph more than once, it is not a function.

Even, Odd, or Neither

• Even: , symmetric about the y-axis. Example:

• Odd: , symmetric about the origin. Example:

• Neither: Does not satisfy either condition.

2. Graph Transformations

Horizontal Shifts

• : shift left by units

• : shift right by units

Vertical Shifts

• : shift up by units

• : shift down by units

Reflections

• : reflect across y-axis

• : reflect across x-axis

Stretching & Shrinking

• Vertical: stretches if , shrinks if

• Horizontal: compresses/stretches in x-direction

Examples

• Basic: , a V-shape with vertex at

• General: , vertex at

• Example: has vertex at

4. Intercepts

X-Intercepts

• Solve

Y-Intercept

• Evaluate

Example

• For , x-intercepts: solve (), y-intercept:

5. Local & Absolute Extrema

Definitions

• Local max: Highest point nearby

• Local min: Lowest point nearby

• Absolute max: Highest point overall

• Absolute min: Lowest point overall

Example

• For , vertex is minimum. No maximum (parabola opens upward).

6. Quadratic Functions

Forms

• Standard:

• Vertex:

Completing the Square (Step by Step)

1. Factor if needed.

2. Add

3. Rewrite as a square.

7. Piecewise Functions

Defined by different rules in different intervals.

• Use open circles (excluded) and closed circles (included).

Example

• is the absolute value function.

8. Symmetry

• Even: Symmetric about y-axis

• Odd: Symmetric about origin

• Neither: No symmetry

9. Increasing & Decreasing Intervals

• Increasing: As increases, goes up

• Decreasing: As increases, goes down

Example

• For , decreasing on , increasing on

10. Average Rate of Change

Formula

This is the slope of the secant line.

11. Secant Lines

A secant line is a line between two points on a curve.

• Equation: where is the slope between two points.

12. Real-World Applications

Population Growth Example

• Formula: Average rate =

• If , , (units per time)

Cost Function Example

• If = cost of producing bicycles

• Average cost per bicycle =

• Solve break-even points by setting

Quick Practice Problems

1. Determine if is even, odd, or neither.

2. Find the vertex and axis of symmetry of .

3. Graph . Identify vertex.

4. For , compute average rate of change from to .

5. A population grows from 0.32g at t = 3.5h to 0.51g at t = 5h. Find average rate of growth.