Transformations Of Cosine Functions: Finding The Equation

Let's dive into how transformations affect trigonometric functions, specifically the cosine function. We're going to break down each transformation step-by-step so you can easily understand how to find the equation of the resulting graph. Guys, this is super useful for exams and understanding graphs in general!

Understanding the Original Function: $y = \cos x$

Before we start messing with transformations, let's make sure we're solid on the original function, $y = \cos x$. This is your basic cosine wave, oscillating between 1 and -1. It starts at a maximum value of 1 when x is 0, goes down to -1 at x = 180°, and completes a full cycle by 360°. Understanding this basic shape is crucial because all transformations are relative to this original curve. Think of it as the foundation upon which we're building our transformed cosine castle. We need to visualize this cosine wave, its peaks and valleys, and its key points to accurately predict how it will change with each transformation.

When we talk about the cosine function, we're dealing with a periodic function. The period of the standard cosine function, $y = \cos x$, is $2\pi$ or 360 degrees. This means the function repeats its values every 360 degrees. Key features of the cosine function include its amplitude (the distance from the midline to the peak or trough), its period, and its phase shift (horizontal shift). Identifying these characteristics in the original function helps us to track how they are altered during the transformations.

Also, keep in mind that the cosine function is an even function, meaning that $\cos(-x) = \cos(x)$. This symmetry about the y-axis can sometimes be helpful when sketching the graph or analyzing its properties. In our case, understanding these basic elements of the cosine function is the first step toward grasping the effects of vertical and horizontal stretches, shifts, and reflections. So, let's keep the original cosine wave in mind as we explore how these transformations modify its shape and position. We want to make sure we have a firm grasp on this foundational function before we proceed to more complex manipulations.

Vertical Stretch by a Factor of 3

First, let's tackle the vertical stretch. When we say the graph is stretched vertically by a factor of 3, it means we're multiplying all the y-values of the original function by 3. So, our new function becomes $y = 3\cos x$. What does this do to the graph? Well, the amplitude, which was originally 1, now becomes 3. This means the graph will now oscillate between 3 and -3 instead of 1 and -1. The peaks are higher, and the valleys are deeper. Essentially, the graph gets taller. A vertical stretch affects the maximum and minimum values of the function, altering its overall appearance. This is a direct scaling along the y-axis, so every point on the original cosine curve is moved farther away from the x-axis by a factor of 3. Remember, the x-values stay the same; only the y-values change.

To visualize this, imagine grabbing the top and bottom of the original cosine wave and pulling them away from each other. The wave becomes elongated vertically. The points where the original cosine function had a y-value of 1 now have a y-value of 3, and the points where it had a y-value of -1 now have a y-value of -3. The x-intercepts, where the y-value is 0, remain unchanged because 3 * 0 = 0. Thus, the vertical stretch primarily impacts the amplitude of the cosine function. Grasping this concept is crucial because it demonstrates how a simple multiplication can drastically change the appearance of the graph.

In summary, the vertical stretch by a factor of 3 transforms the original function $y = \cos x$ into $y = 3\cos x$. This increases the amplitude of the cosine wave from 1 to 3, making the peaks and troughs more pronounced. It's a straightforward transformation, but it has a significant impact on the graph's appearance, making it taller and more dramatic. As we move on to the next transformations, we'll see how these changes accumulate to create the final, transformed graph.

Horizontal Compression by a Factor of $\frac{1}{3}$

Now, let's compress the graph horizontally by a factor of $\frac{1}{3}$. This is a bit trickier than the vertical stretch because it affects the x-values inside the cosine function. To compress the graph horizontally by a factor of $\frac{1}{3}$, we replace x with 3x. So, our function now becomes $y = 3\cos(3x)$. What does this do? It changes the period of the function. The original period of $\cos x$ is 360°. When we have $\cos(3x)$, the new period is $\frac{360^\[\circ]}{3} = 120^\[\circ]$. This means the graph completes one full cycle in just 120° instead of 360°. The graph is squished horizontally.

Imagine pushing the cosine wave from both sides, compressing it into a smaller space. The peaks and troughs now occur more frequently. The x-intercepts also shift closer together. This horizontal compression is effectively speeding up the cosine function. Instead of taking 360 degrees to complete one cycle, it now completes it in 120 degrees. The amplitude remains the same (3 in this case), but the frequency of the oscillations increases. This transformation might be a little mind-bending, but visualizing it as squishing the graph horizontally can help.

Understanding this horizontal compression is essential because it illustrates how changing the argument of the cosine function (the part inside the cosine) affects its period. By multiplying x by 3, we are essentially speeding up the function, causing it to repeat its cycle more quickly. This transformation contrasts with the vertical stretch, which affected the amplitude but left the period unchanged. The combination of vertical and horizontal transformations allows us to manipulate the shape and frequency of the cosine wave in various ways, leading to a wide range of possible graphs. So, let's keep these concepts in mind as we proceed to the next steps, where we'll shift the graph both vertically and horizontally.

Downward Shift of 1 Unit

Next up, we shift the graph downward by 1 unit. This is a simple vertical shift. To shift the entire graph down by 1 unit, we subtract 1 from the function. So, our function becomes $y = 3\cos(3x) - 1$. What does this do? It moves the entire graph down along the y-axis. The midline of the graph, which was previously at y = 0, now moves to y = -1. The graph now oscillates between 2 and -4 instead of 3 and -3.

Imagine taking the entire cosine wave and sliding it down one unit on the y-axis. Every point on the graph moves down by the same amount. The peaks, troughs, and x-intercepts all shift downwards. This transformation does not change the shape or period of the wave; it simply repositions it on the coordinate plane. The downward shift affects the vertical position of the entire graph, changing its relationship to the x-axis. This is a straightforward but important transformation that can drastically alter the appearance of the graph and its key features.

To visualize this, think of the x-axis as a reference line. Before the shift, the cosine wave oscillated around this line. After the shift, the cosine wave oscillates around the line y = -1. This change in the midline is a key indicator of a vertical shift. Understanding this transformation is essential for accurately sketching and interpreting cosine graphs, as it helps us to understand how vertical positioning affects the overall appearance of the function. So, let's keep this downward shift in mind as we move on to the final transformation, which involves a horizontal shift.

Leftward Shift of $45^\[\circ]$

Finally, let's shift the graph to the left by 45°. This is a horizontal shift, also known as a phase shift. To shift the graph to the left by 45°, we replace x with (x + 45°) inside the cosine function. Our function now becomes $y = 3\cos(3(x + 45^\[\circ])) - 1$. What does this do? It moves the entire graph to the left along the x-axis. The graph now starts its cycle 45° earlier.

Imagine taking the cosine wave and sliding it to the left along the x-axis. Every point on the graph moves to the left by the same amount. The peaks, troughs, and x-intercepts all shift horizontally. This transformation does not change the shape or period of the wave; it simply repositions it on the coordinate plane. The horizontal shift affects the starting point of the cosine function, changing its phase relative to the y-axis.

Important Note: Be careful with the horizontal compression! The horizontal shift happens after the compression. That's why we have 3(x + 45°). This means the 45° shift is affected by the compression as well. It's not a simple 45° shift; it's compressed along with the x-axis. To see the actual shift, we need to distribute the 3: $y = 3\cos(3x + 135^\[\circ]) - 1$. So, the actual leftward shift is 135° after considering the compression.

The Final Equation

Putting it all together, the final equation of the transformed graph is: $y = 3\cos(3(x + 45^\[\circ])) - 1$. This equation represents a cosine function that has been stretched vertically by a factor of 3, compressed horizontally by a factor of $\frac{1}{3}$, shifted down by 1 unit, and shifted to the left by 45° (after considering the horizontal compression).

Therefore, rewriting the equation

$y = 3\cos(3(x + 45^\[\circ])) - 1$

$y = 3\cos(3x + 135^\[\circ]) - 1$

Conclusion

So, remember guys, when transforming trigonometric functions, take it one step at a time. Vertical stretches affect the amplitude, horizontal compressions affect the period, and shifts move the entire graph. Combining these transformations allows you to create a wide variety of cosine functions. And always double-check the order of operations, especially when dealing with horizontal compressions and shifts!