Select The Correct Answer.The Parent Cosine Function Is Shifted 5 Units Left, Then Vertically Stretched By A Factor Of 4, And Shifted Up 2 Units. Which Statement Is True About The Graph Of The Transformed Function?A. The Midline Is $y = 4$.B.

Feb 27, 2025 by ADMIN 243 views

**Transforming the Parent Cosine Function**

The parent cosine function, denoted as $y = \cos(x)$ , is a fundamental function in mathematics that has numerous applications in various fields. When we transform this function, we can create new functions that exhibit different characteristics. In this article, we will explore the transformation of the parent cosine function by shifting it 5 units left, vertically stretching it by a factor of 4, and shifting it up 2 units.

Understanding the Transformation

To understand the transformation of the parent cosine function, let's break down the individual steps involved.

Shifting the Function 5 Units Left

When we shift the function 5 units left, the new function becomes $y = \cos(x + 5)$ . This means that the graph of the function will be shifted 5 units to the left, resulting in a change in the x-coordinates of the points on the graph.

Vertically Stretching the Function by a Factor of 4

After shifting the function 5 units left, we vertically stretch it by a factor of 4. This means that the new function becomes $y = 4\cos(x + 5)$ . The vertical stretching will result in a change in the y-coordinates of the points on the graph, making the graph more compressed or stretched.

Shifting the Function Up 2 Units

Finally, we shift the function up 2 units, resulting in the new function $y = 4\cos(x + 5) + 2$ . This means that the graph of the function will be shifted 2 units upwards, resulting in a change in the y-coordinates of the points on the graph.

Analyzing the Graph of the Transformed Function

Now that we have transformed the parent cosine function, let's analyze the graph of the transformed function. The graph of the transformed function will exhibit the following characteristics:

Midline: The midline of the graph of the transformed function is the horizontal line that passes through the midpoint of the graph. Since the function is shifted up 2 units, the midline will be at $y = 2$ .
Amplitude: The amplitude of the graph of the transformed function is the distance from the midline to the maximum or minimum point on the graph. Since the function is vertically stretched by a factor of 4, the amplitude will be 4 units.
Period: The period of the graph of the transformed function is the distance between two consecutive points on the graph that have the same y-coordinate. Since the function is shifted 5 units left, the period will remain the same as the parent cosine function, which is $2\pi$ .

Conclusion

In conclusion, the graph of the transformed function will have a midline at $y = 2$ , an amplitude of 4 units, and a period of $2\pi$ . Therefore, the correct statement about the graph of the transformed function is:

A. The midline is $y = 2$

This statement is true because the graph of the transformed function is shifted up 2 units, resulting in a midline at $y = 2$ . The other options are incorrect because the midline is not at $y = 4$ (option B), and the amplitude is not 1 unit (option C).

Final Answer

In our previous article, we explored the transformation of the parent cosine function by shifting it 5 units left, vertically stretching it by a factor of 4, and shifting it up 2 units. We analyzed the graph of the transformed function and determined that the midline is at $y = 2$ , the amplitude is 4 units, and the period is $2\pi$ . In this article, we will answer some frequently asked questions about transforming the parent cosine function.

Q: What is the effect of shifting the parent cosine function 5 units left?

A: When we shift the parent cosine function 5 units left, the new function becomes $y = \cos(x + 5)$ . This means that the graph of the function will be shifted 5 units to the left, resulting in a change in the x-coordinates of the points on the graph.

Q: What is the effect of vertically stretching the parent cosine function by a factor of 4?

A: When we vertically stretch the parent cosine function by a factor of 4, the new function becomes $y = 4\cos(x + 5)$ . This means that the graph of the function will be stretched vertically by a factor of 4, resulting in a change in the y-coordinates of the points on the graph.

Q: What is the effect of shifting the parent cosine function up 2 units?

A: When we shift the parent cosine function up 2 units, the new function becomes $y = 4\cos(x + 5) + 2$ . This means that the graph of the function will be shifted 2 units upwards, resulting in a change in the y-coordinates of the points on the graph.

Q: What is the midline of the graph of the transformed function?

A: The midline of the graph of the transformed function is the horizontal line that passes through the midpoint of the graph. Since the function is shifted up 2 units, the midline will be at $y = 2$ .

Q: What is the amplitude of the graph of the transformed function?

A: The amplitude of the graph of the transformed function is the distance from the midline to the maximum or minimum point on the graph. Since the function is vertically stretched by a factor of 4, the amplitude will be 4 units.

Q: What is the period of the graph of the transformed function?

A: The period of the graph of the transformed function is the distance between two consecutive points on the graph that have the same y-coordinate. Since the function is shifted 5 units left, the period will remain the same as the parent cosine function, which is $2\pi$ .

Q: How do I determine the midline, amplitude, and period of the graph of a transformed function?

A: To determine the midline, amplitude, and period of the graph of a transformed function, you need to analyze the individual transformations that were applied to the parent function. For example, if the function is shifted up 2 units, the midline will be at $y = 2$ . If the function is vertically stretched by a factor of 4, the amplitude will be 4 units. If the function is shifted 5 units left, the period will remain the same as the parent function.

Conclusion

In conclusion, transforming the parent cosine function by shifting it 5 units left, vertically stretching it by a factor of 4, and shifting it up 2 units results in a graph with a midline at $y = 2$ , an amplitude of 4 units, and a period of $2\pi$ . By analyzing the individual transformations that were applied to the parent function, you can determine the midline, amplitude, and period of the graph of the transformed function.

Final Answer

The final answer is:

Midline: $y = 2$
Amplitude: 4 units
Period: $2\pi$