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, the relation 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 right by units.

• : shift left by units.

Vertical Shifts

• : shift up by units.

• : shift down by units.

Reflections

• : reflect across the y-axis.

• : reflect across the x-axis.

Stretching & Shrinking

• Vertical: : stretch if , shrink if .

• Horizontal: : compression/stretch in x-direction.

• Basic: , a V-shape with vertex at .

• General: , vertex at .

• Example: has vertex at .

4. Intercepts

X-Intercepts

• Solve .

Y-Intercept

• Evaluate .

Example: 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: 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) on graphs.

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: 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 , , then (units per time)

Cost Function Example

• If is the cost of producing bicycles, average cost per bicycle is .

• 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 the vertex.

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

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