Shifting, Reflecting, Stretching, and Compressing Graphs

Introduction

Understanding how the graph of a function changes under various transformations is a fundamental skill in precalculus. Transformations include shifting (translations), reflecting, stretching, and compressing graphs. Mastery of these concepts allows for quick graph sketching and deeper insight into function behavior.

Vertical Shifting

Definition and Properties

• Vertical shifting moves a graph up or down without changing its shape.

• The general form is:

• If c is positive, the graph shifts up by c units.

• If c is negative, the graph shifts down by units.

Example: Starting with (blue graph):

• (red graph) shifts the parabola up by 2 units.

• (blue graph) shifts the parabola down by 1 unit.

Horizontal Shifting

Definition and Properties

• Horizontal shifting moves a graph left or right.

• The general form is:

• To determine the shift, solve for .

• If c is positive, the graph shifts left by c units.

• If c is negative, the graph shifts right by units.

Example: Starting with (red graph):

• (blue graph) shifts the parabola left by 2 units.

• (blue graph) shifts the parabola right by 1 unit.

Reflecting Graphs

Reflection with Respect to the x-axis

• The general form is:

• This transformation reflects the graph over the x-axis, changing the sign of all y-values.

Example: (red graph), (blue graph) is the reflection over the x-axis.

Reflection with Respect to the y-axis

• The general form is:

• This transformation reflects the graph over the y-axis, changing the sign of all x-values.

Example: (red graph), (green graph) is the reflection over the y-axis.

Stretching and Compressing Graphs

Vertical Stretching and Compressing

• The general form is:

• If , the graph stretches vertically by a factor of (moves farther from the x-axis).

• If , the graph compresses vertically by a factor of (moves closer to the x-axis).

Examples:

• (red graph), (blue graph) is a vertical stretch by 2.

• (blue graph) is a vertical compression by .

Horizontal Stretching and Compressing

• The general form is:

• If , the graph compresses horizontally by a factor of (moves closer to the y-axis).

• If , the graph stretches horizontally by a factor of (moves farther from the y-axis).

Examples:

• (red graph), (blue graph) is a horizontal compression by .

• (blue graph) is a horizontal stretch by 2.

Combined Transformations

Order of Transformations

• When multiple transformations are applied, the order can affect the final graph.

• Typical order: Horizontal shifts → Stretches/Compressions → Reflections → Vertical shifts.

Example: To shift two units up and three units to the left:

• First, shift left:

• Then, shift up:

Worked Examples and Applications

• Example 1: If you reflect over the x-axis, stretch vertically by 3, then translate 7 units up, the resulting function is .

• Example 2: Given , shift left 3, reflect over the y-axis, stretch vertically by 4, then translate up 5: .

• Example 3: If you shift three units to the left and one unit up: .

Summary Table: Transformations of Functions

Practice Problems

• Given , what is the equation if the graph is shifted 2 units up and 3 units to the left? Answer:

• If is reflected over the x-axis, stretched vertically by 3, and translated 7 units up, what is the new equation? Answer:

• Given , shift left 3, reflect over the y-axis, stretch vertically by 4, then translate up 5: Answer:

• If you shift three units to the left and one unit up: Answer:

Additional info: The above notes synthesize and expand upon the provided slides and practice problems, ensuring all key transformation types are covered with definitions, formulas, and examples.