Which Equation Represents The Graph Of $y=\cos X$ Flipped Across The \$x$-axis$ And Then Shifted Vertically Up By 2 Units?A. $y=\cos (x-2)$ B. $y=-\cos X-2$ C. \$y=-\cos X+2$[/tex\]
Introduction
In mathematics, transforming trigonometric graphs is an essential concept that helps us understand and manipulate various functions. One of the fundamental transformations is flipping a graph across the x-axis and shifting it vertically. In this article, we will explore how to represent the graph of y = cos(x) flipped across the x-axis and then shifted vertically up by 2 units.
Understanding the Original Graph
The original graph of y = cos(x) is a periodic function that oscillates between -1 and 1. The graph has a period of 2Ï€, which means it repeats every 2Ï€ units. The graph is symmetric about the y-axis and has a maximum value of 1 and a minimum value of -1.
Flipping the Graph Across the X-Axis
To flip the graph of y = cos(x) across the x-axis, we need to multiply the function by -1. This will reflect the graph about the x-axis, resulting in a new function:
y = -cos(x)
The graph of y = -cos(x) is the same as the original graph of y = cos(x) but reflected about the x-axis. The graph now oscillates between 1 and -1, with a maximum value of 1 and a minimum value of -1.
Shifting the Graph Vertically Up by 2 Units
To shift the graph of y = -cos(x) vertically up by 2 units, we need to add 2 to the function. This will move the graph up by 2 units, resulting in a new function:
y = -cos(x) + 2
The graph of y = -cos(x) + 2 is the same as the graph of y = -cos(x) but shifted up by 2 units. The graph now oscillates between 3 and 1, with a maximum value of 3 and a minimum value of 1.
Conclusion
In conclusion, to represent the graph of y = cos(x) flipped across the x-axis and then shifted vertically up by 2 units, we need to multiply the function by -1 and then add 2. The resulting function is:
y = -cos(x) + 2
This function represents the graph of y = cos(x) flipped across the x-axis and then shifted vertically up by 2 units.
Answer
The correct answer is:
C. y = -cos x + 2
Explanation
The correct answer is C. y = -cos x + 2 because it represents the graph of y = cos(x) flipped across the x-axis and then shifted vertically up by 2 units. The function y = -cos x + 2 is the result of multiplying the original function by -1 and then adding 2.
Comparison with Other Options
Let's compare the correct answer with the other options:
A. y = cos (x-2)
This option is incorrect because it represents the graph of y = cos(x) shifted horizontally to the right by 2 units, not flipped across the x-axis and shifted vertically up by 2 units.
B. y = -cos x - 2
This option is incorrect because it represents the graph of y = cos(x) flipped across the x-axis and then shifted vertically down by 2 units, not up by 2 units.
Final Thoughts
Q&A: Transforming Trigonometric Graphs
Q: What is the difference between flipping a graph across the x-axis and shifting it vertically? A: Flipping a graph across the x-axis involves multiplying the function by -1, resulting in a reflection of the graph about the x-axis. Shifting a graph vertically involves adding or subtracting a value from the function, resulting in a movement of the graph up or down.
Q: How do I flip the graph of y = sin(x) across the x-axis? A: To flip the graph of y = sin(x) across the x-axis, you need to multiply the function by -1. The resulting function is y = -sin(x).
Q: How do I shift the graph of y = cos(x) vertically up by 3 units? A: To shift the graph of y = cos(x) vertically up by 3 units, you need to add 3 to the function. The resulting function is y = cos(x) + 3.
Q: What is the effect of shifting a graph horizontally? A: Shifting a graph horizontally involves adding or subtracting a value from the function, resulting in a movement of the graph left or right. For example, shifting the graph of y = sin(x) to the right by 2 units results in the function y = sin(x - 2).
Q: How do I flip the graph of y = tan(x) across the x-axis and then shift it vertically up by 2 units? A: To flip the graph of y = tan(x) across the x-axis, you need to multiply the function by -1. The resulting function is y = -tan(x). To shift the graph vertically up by 2 units, you need to add 2 to the function. The resulting function is y = -tan(x) + 2.
Q: What is the difference between a periodic function and a non-periodic function? A: A periodic function is a function that repeats its values at regular intervals, known as the period. A non-periodic function is a function that does not repeat its values at regular intervals.
Q: How do I determine the period of a trigonometric function? A: The period of a trigonometric function can be determined by the coefficient of the variable in the function. For example, the period of the function y = sin(x) is 2π, while the period of the function y = sin(2x) is π.
Q: What is the effect of multiplying a trigonometric function by a constant? A: Multiplying a trigonometric function by a constant results in a vertical stretch or compression of the graph. For example, multiplying the function y = sin(x) by 2 results in the function y = 2sin(x), which is a vertical stretch of the original graph.
Q: How do I graph a trigonometric function using a calculator? A: To graph a trigonometric function using a calculator, you need to enter the function in the calculator's graphing mode and adjust the window settings as needed to view the graph.
Q: What are some common applications of trigonometric functions? A: Trigonometric functions have many applications in various fields, including physics, engineering, and computer science. Some common applications include modeling periodic phenomena, such as sound waves and light waves, and solving problems involving right triangles and circular motion.
Conclusion
Transforming trigonometric graphs is an essential concept in mathematics that helps us understand and manipulate various functions. By flipping and shifting graphs, we can create new functions that represent different scenarios. In this article, we explored some common questions and answers related to transforming trigonometric graphs, including flipping and shifting, periodic and non-periodic functions, and graphing using a calculator.