Linear Transformations Vertical and Horizontal Stretching and Compressing Examples

Keith Mann

29 Oct 201308:43

Summary

TLDRThis video explains how to horizontally and vertically stretch functions. It walks through examples, first demonstrating how to horizontally stretch a function by a factor of two and discussing the effect on the slope, which decreases. The presenter also shows how to graph the transformations and update the function’s equation. Next, the video covers vertical stretching by a factor of two, explaining how to multiply the entire function by the stretch factor, and showing both the graphing and notation methods to find the new function equation.

Takeaways

🧠 Horizontally stretching a function by a factor of two reduces the slope.
🔢 Horizontal stretching affects only the X portion of a function.
🔀 You can either divide the X by 2 or multiply it by 1/2 for horizontal stretching.
📉 After horizontally stretching, the slope of a line decreases (e.g., from 3 to 3/2).
📝 Graphing helps visualize how stretching changes the shape of a function.
📈 When stretching horizontally, multiply each X-coordinate by the stretch factor.
📊 Vertically stretching by a factor of two increases the slope and makes the function taller.
⬆️ Vertically stretching changes the Y-coordinates of the function.
✏️ Multiplying the entire function by the stretch factor applies vertical stretching.
📐 Using both graphing and notation helps reinforce understanding of function transformation.

Q & A

What is the main focus of the video?
-The video focuses on stretching and compressing functions, specifically discussing how to horizontally and vertically stretch a function and how to write the new equation after these transformations.
What happens when a function is horizontally stretched?
-When a function is horizontally stretched, the slope of the function decreases, making the function appear 'flatter.' The x-values are affected, and this results in the graph stretching outward horizontally.
How do you horizontally stretch a function by a factor of 2?
-To horizontally stretch a function by a factor of 2, you need to divide the x-variable by 2. Alternatively, you can multiply the x-variable by 1/2, as both approaches yield the same effect.
What impact does horizontal stretching have on the slope of the function?
-Horizontal stretching reduces the slope of the function. For example, if the slope is initially 3, it becomes 3/2 after a horizontal stretch by a factor of 2.
What does the notation 'G(x)' represent in the video?
-In the video, 'G(x)' represents the transformed function after applying either a horizontal or vertical stretch to the original function 'f(x).'
How can you visually check if a horizontal stretch was applied correctly?
-You can visually check by graphing both the original function and the transformed function. The transformed function should look stretched horizontally, with the x-coordinates of key points multiplied by the stretching factor.
What is the key difference between horizontal and vertical stretching?
-Horizontal stretching affects the x-values of the function and makes the graph wider, while vertical stretching affects the y-values, making the graph taller or shorter depending on the factor.
How do you vertically stretch a function by a factor of 2?
-To vertically stretch a function by a factor of 2, you multiply the entire function by 2, which scales the y-values by 2 and makes the graph taller.
What happens to the slope when a function is vertically stretched?
-When a function is vertically stretched, the slope increases. For example, if the initial slope is 3, it becomes 6 after a vertical stretch by a factor of 2.
What are two methods mentioned in the video for finding the transformed function?
-The video explains two methods: using notation (mathematically altering the function) and graphing (visually transforming the function by adjusting the coordinates and then plotting the new points).

Outlines

00:00

📐 Understanding Horizontal Stretching of Functions

In this section, the speaker introduces the concept of stretching and compressing functions, with a focus on horizontally stretching a function by a factor of two. The key idea is that when horizontally stretching, the slope of the function decreases, and this transformation affects only the X-coordinate. The speaker explains how to represent this change in notation and graphically, emphasizing that multiplying the X-coordinate by 1/2 has the same effect as dividing by 2. They illustrate this process by modifying the function and graphing the changes, showing how the slope changes from 3 to 3/2.

05:00

🖊 Visualizing Horizontal Stretch through Graphing

The speaker continues the explanation by graphing the original function and its horizontally stretched version. They show how to adjust the X-coordinates by multiplying them by 2 to reflect the horizontal stretch. By comparing the original and transformed functions, they demonstrate how the graph visually represents the horizontal stretching, with key points shifting on the X-axis. The speaker notes that the Y-intercept remains the same while the slope changes, reaffirming that the slope decreases as the function stretches horizontally.

📏 Vertically Stretching Functions by a Factor of Two

In this section, the focus shifts to vertical stretching, where the speaker describes stretching a function vertically by a factor of two. They graph the original function, and then explain how vertical stretching affects the Y-coordinates. By multiplying the Y-values by 2, the graph becomes taller, with steeper slopes and more negative Y-intercepts. The speaker shows that a vertical stretch increases both the slope and Y-intercept, changing the equation from 3x - 2 to 6x - 4.

📊 Applying Vertical Stretching Using Notation

To reinforce the concept, the speaker demonstrates how to apply vertical stretching through notation. They explain that vertically stretching a function means multiplying the entire function by a constant factor, in this case, by 2. This transformation results in multiplying both the slope and the Y-intercept by 2, leading to a new equation for the transformed function. The speaker concludes by summarizing the process of vertical stretching and showing that it can be understood both graphically and through algebraic manipulation.

Mindmap

Keywords

💡Horizontally Stretch

Horizontally stretching a function refers to expanding the function along the x-axis, making it appear wider. In the video, the instructor explains how multiplying the x-values by a factor (in this case, by 2) affects the slope, making it smaller. This transformation only changes the x-coordinates of the function's graph, as demonstrated when the slope of the function decreases after the horizontal stretch.

💡Vertically Stretch

A vertical stretch affects the y-values of a function, making it taller along the y-axis. In the video, the instructor describes vertically stretching by a factor of 2, where the y-coordinates are multiplied, leading to a steeper slope. The example shows how this transformation changes the height of the graph, making the line taller, with both the slope and the y-intercept becoming larger in magnitude.

💡Slope

The slope of a function represents its rate of change, or how steep the graph is. The instructor emphasizes that horizontal and vertical stretches change the slope of the function. For example, a horizontal stretch decreases the slope, while a vertical stretch increases it, as seen when the slope goes from 3 to 3/2 after the horizontal stretch.

💡Factor of Two

The phrase 'factor of two' refers to the multiplier used to stretch or compress the function. A factor of two can either horizontally stretch or vertically stretch a function, depending on whether it is applied to the x-values or y-values. In the script, the factor of two is used to stretch functions in both horizontal and vertical directions in different examples.

💡Transformation

A transformation refers to any operation that changes the position or shape of a graph. In the video, transformations include horizontal and vertical stretches, where the function's equation and graph are modified by factors applied to either the x-values or y-values. The transformation alters the appearance of the graph without changing its fundamental structure.

💡X-coordinate

The x-coordinate represents the horizontal position of a point on the graph. In the video, the instructor explains that horizontal stretches only affect the x-coordinates, such as when multiplying them by 2, which causes the graph to spread out horizontally. This change influences the appearance of the function, but not the y-values.

💡Y-coordinate

The y-coordinate represents the vertical position of a point on a graph. In the context of vertical stretches, the instructor demonstrates how multiplying the y-coordinates by a factor of two stretches the function upward, making it appear taller. This transformation only impacts the y-values, altering the height and steepness of the graph.

💡Graphing

Graphing is the visual representation of a function on a coordinate plane. The instructor in the video shows how to graph both the original function and its transformations, helping to visualize how stretches affect the function’s shape. This process involves plotting points based on their x and y coordinates and drawing the corresponding lines.

💡Notation

Notation in mathematics refers to the symbols and structure used to represent functions and transformations. In the video, the instructor first demonstrates the use of notation to represent transformations like horizontal and vertical stretches. For example, G(x) is used to denote the transformed function derived from the original function f(x).

💡Y-intercept

The y-intercept is the point where a function's graph crosses the y-axis. In the video, the y-intercept changes after transformations like vertical stretches, as multiplying the y-values affects its position. For example, the original y-intercept of -2 changes to -4 after a vertical stretch by a factor of two.

Highlights

Introduction to stretching and compressing functions, focusing on both horizontal and vertical transformations.

Horizontal stretches affect only the x-coordinates of a function, causing a decrease in slope as the function appears 'stretched out'.

Key concept: Horizontally stretching a function results in the slope decreasing, as shown by graph transformations.

Demonstrating that to horizontally stretch by a factor of two, the x-portion of the function is divided by two or multiplied by 1/2.

When transforming a function horizontally, the x-coordinates of points are multiplied by the stretch factor to plot the new function.

Graphing approach: Horizontal stretching by multiplying the x-coordinates of specific points on the graph (e.g., 2 becomes 4, 1 becomes 2).

Visual demonstration of horizontal stretch: Observing how the function's slope decreases when plotted after the transformation.

Comparison of initial and transformed functions using graphing techniques to showcase the changes in slope and shape.

Vertical stretches affect the y-coordinates, leading to the function becoming 'taller' and the slope increasing.

In a vertical stretch, the y-coordinates of the function are multiplied by the stretch factor (e.g., multiplying by 2).

Graphing approach for vertical stretching: Observing how specific y-values (e.g., 1 becomes 2, -2 becomes -4) shift after transformation.

Visual demonstration of vertical stretch: The function grows taller, the y-intercept shifts, and the slope increases.

Key takeaway: Vertically stretching by a factor of two doubles the slope and y-intercept of the function.

Vertical stretches require multiplying the entire function by the stretch factor to get the new transformed equation.

Conclusion: Two methods to handle stretching transformations—graphing and using mathematical notation—both yield the same results.