Graphing And Analyzing Trigonometric Functions F(x) = Tan(x) And G(x) = 2sin(x) + 1
Hey guys! Today, we're diving deep into the fascinating world of trigonometric functions, specifically focusing on $f(x) = \tan(x)$ and $g(x) = 2\sin(x) + 1$. We'll be sketching their graphs, figuring out their ranges and periods, and tackling some interesting questions along the way. So, buckle up and let's get started!
a. Sketching the Graphs of f(x) = tan(x) and g(x) = 2sin(x) + 1
Let's kick things off by sketching the graphs of these two trigonometric functions on the same set of axes. This will give us a visual representation of their behavior and help us understand their key characteristics. The domain we're working with is $-30 \leqslant x \leqslant 360$ degrees, so we need to make sure our graphs cover this interval.
Understanding the Tangent Function f(x) = tan(x)
First up, we have the tangent function, $f(x) = \tan(x)$. Now, the tangent function is a bit of a wild child compared to sine and cosine. It's defined as the ratio of sine to cosine, $ an(x) = \frac{\sin(x)}{\cos(x)}$. This definition is crucial because it tells us that tangent will be undefined whenever cosine is zero. Cosine is zero at 90 degrees and 270 degrees (and their coterminal angles), which means we'll have vertical asymptotes at these points on the graph of $f(x) = \tan(x)$. These asymptotes are like invisible walls that the graph approaches but never touches.
The tangent function has a period of 180 degrees, which means its pattern repeats every 180 degrees. In the interval from -30 to 360 degrees, we'll see this pattern repeating multiple times. Between the asymptotes, the tangent function increases from negative infinity to positive infinity. It passes through zero at angles where sine is zero (0 degrees, 180 degrees, 360 degrees). So, when you're sketching the graph, remember those vertical asymptotes and the increasing behavior between them.
To get a more accurate sketch, it's helpful to plot a few key points. We know $ an(0) = 0$, $ an(45) = 1$, and $ an(-45) = -1$. Using these points as guides, we can draw the characteristic curves of the tangent function between the asymptotes. Remember that the graph gets increasingly steep as it approaches the asymptotes.
Unveiling the Transformed Sine Function g(x) = 2sin(x) + 1
Next, we have $g(x) = 2\sin(x) + 1$. This function is a transformation of the basic sine function, $\sin(x)$. Let's break down these transformations step by step. The '2' in front of the sine function represents a vertical stretch by a factor of 2. This means the amplitude of the graph, which is the distance from the midline to the peak or trough, is now 2 instead of 1. In simpler terms, the graph will be twice as tall as the regular sine wave.
The '+ 1' at the end represents a vertical shift upwards by 1 unit. This means the entire graph is lifted 1 unit higher on the y-axis. The midline of the graph, which is the horizontal line that runs through the middle of the wave, is now at $y = 1$ instead of $y = 0$. These transformations significantly impact the graph's position and shape, but the fundamental wave-like nature of the sine function remains. Understanding these transformations is key to accurately sketching the graph.
The sine function, $\sin(x)$, oscillates between -1 and 1. With the vertical stretch and shift, $g(x)$ will oscillate between $-1 * 2 + 1 = -1$ and $1 * 2 + 1 = 3$. The graph will cross the midline (y = 1) at the same x-values where the regular sine function crosses the x-axis (0 degrees, 180 degrees, 360 degrees). The peaks of the wave will occur at 90 degrees plus multiples of 360 degrees, and the troughs will occur at 270 degrees plus multiples of 360 degrees.
Putting it All Together on the Same Axes
Now, the real fun begins! We'll sketch both graphs, $f(x) = \tan(x)$ and $g(x) = 2\sin(x) + 1$, on the same set of axes. This will allow us to visualize their relationship and identify any points of intersection. Start by drawing the vertical asymptotes for the tangent function at 90 degrees, 270 degrees, and their coterminal angles within our domain. Then, sketch the characteristic curves of the tangent function between these asymptotes.
Next, sketch the graph of $g(x) = 2\sin(x) + 1$. Remember to account for the vertical stretch and shift. The graph should oscillate between -1 and 3, with a midline at $y = 1$. By plotting a few key points and connecting them smoothly, you'll get a clear picture of the sine wave. With both graphs on the same axes, you can visually identify their ranges, periods, and any intersection points.
b. Determining the Range of g(x) = 2sin(x) + 1
The range of a function is the set of all possible output values (y-values) that the function can produce. For $g(x) = 2\sin(x) + 1$, we've already discussed how the vertical stretch and shift affect the graph. The basic sine function, $\sin(x)$, has a range of [-1, 1]. This means its output values are always between -1 and 1, inclusive. The transformations we've applied to create $g(x)$ change this range.
As we mentioned earlier, the '2' in front of the sine function stretches the graph vertically by a factor of 2. This means the range becomes [-2, 2]. Then, the '+ 1' shifts the entire graph upwards by 1 unit. This means we add 1 to both the minimum and maximum values of the range. So, the range of $g(x) = 2\sin(x) + 1$ is [-2 + 1, 2 + 1], which simplifies to [-1, 3]. Therefore, the range of g(x) is all real numbers between -1 and 3, inclusive.
In other words, the graph of g(x) will never go below y = -1 or above y = 3. Visually, you can confirm this by looking at the sketch of the graph. The lowest point on the graph is at y = -1, and the highest point is at y = 3. This is a direct consequence of the transformations we applied to the basic sine function. To summarize, the range of g(x) is crucial for understanding the function's behavior and its limitations in terms of output values.
c. Finding the Period of f(x) = tan(x)
The period of a trigonometric function is the length of one complete cycle before the pattern repeats itself. For the tangent function, $f(x) = \tan(x)$, the period is different from that of sine and cosine. The period of the tangent function is a fundamental property that distinguishes it from other trigonometric functions. Understanding the period helps us predict the function's behavior over extended intervals and is essential for sketching its graph accurately.
The basic sine and cosine functions have a period of 360 degrees, or $2\pi$ radians. This means their patterns repeat every 360 degrees. However, the tangent function has a period of 180 degrees, or $\pi$ radians. This is because the tangent function's pattern repeats after every vertical asymptote. Remember that tangent is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$, and its asymptotes occur when $\cos(x) = 0$. Cosine is zero at 90 degrees and 270 degrees, which are 180 degrees apart. This 180-degree interval is the period of the tangent function.
Visually, you can see the 180-degree periodicity on the graph of $f(x) = \tan(x)$. The characteristic curve of the tangent function, which increases from negative infinity to positive infinity between the asymptotes, repeats every 180 degrees. This means that if you were to shift the graph 180 degrees to the left or right, it would perfectly overlap with the original graph. In conclusion, the period of f(x) = tan(x) is 180 degrees.
d. Discussion
Let's recap what we've learned about these trigonometric functions. We've successfully sketched the graphs of $f(x) = \tan(x)$ and $g(x) = 2\sin(x) + 1$ on the same set of axes. This visual representation allowed us to understand their behavior, identify key features like asymptotes, and compare their patterns. We also determined the range of g(x) to be [-1, 3], meaning the function's output values are limited to this interval. Moreover, we found that the period of f(x) = tan(x) is 180 degrees, which is a crucial characteristic of the tangent function.
Understanding the transformations applied to trigonometric functions is crucial. For $g(x) = 2\sin(x) + 1$, the vertical stretch and shift significantly altered the range but didn't affect the period. The period remains 360 degrees, just like the basic sine function. However, the range of the transformed function is different, reflecting the vertical adjustments. By analyzing the transformations, we can accurately predict how the graph will change compared to the parent function.
Comparing the graphs of tangent and sine functions highlights their distinct properties. Tangent has vertical asymptotes and a period of 180 degrees, while sine oscillates smoothly with a period of 360 degrees. These differences arise from their definitions and the behavior of sine and cosine. Tangent's behavior near its asymptotes is particularly noteworthy, as it approaches infinity, unlike the bounded sine function. These fundamental differences lead to diverse applications in mathematics and physics, where each function models different phenomena.
Further exploration could involve finding the points of intersection between the two graphs. This would require solving the equation $\tan(x) = 2\sin(x) + 1$. However, this equation is transcendental and generally requires numerical methods to find the solutions. Graphically, the intersection points represent the x-values where the two functions have the same y-value. Identifying these points gives a deeper understanding of the relationship between the functions and their values.
Overall, we've covered significant ground in understanding these trigonometric functions. Sketching the graphs, finding the range, and determining the period are fundamental skills in trigonometry. These concepts provide a solid foundation for more advanced topics like trigonometric identities, equations, and applications in various fields. Remember, practice makes perfect, so keep exploring and experimenting with different trigonometric functions to deepen your understanding!