Graphing Cosine Functions Understanding G(x) = Cos(x - 45 Degrees)

by ADMIN 67 views

Sketch the graph of g(x) = cos(x - 45°) for x in [0°, 360°], showing intercepts and turning points. Find the values of x in the given interval where g(x) satisfies certain conditions (not specified).

Graphing cosine functions can seem daunting at first, but with a systematic approach and a clear understanding of the transformations involved, it becomes a manageable task. This article delves into the process of sketching the graph of a specific cosine function, $g(x) = \cos(x - 45^{\circ})$, within a given interval, and determining its key features such as intercepts and turning points. We will also explore how to find the values of $x$ within the specified interval where the function satisfies certain conditions. This comprehensive guide aims to equip you with the knowledge and skills necessary to confidently graph and analyze cosine functions.

Understanding the Basic Cosine Function

Before we dive into the specifics of $g(x) = \cos(x - 45^{\circ})$, let's first revisit the fundamental properties of the basic cosine function, $f(x) = \cos(x)$. The cosine function is a periodic function with a period of $360^{\circ}$ or $2\pi$ radians. This means that the graph repeats itself every $360^{\circ}$. The amplitude of the basic cosine function is 1, which means the function oscillates between -1 and 1. The key points of the cosine function within one period (from $0^{\circ}$ to $360^{\circ}$) are:

  • (0∘,1)(0^{\circ}, 1)

  • (90∘,0)(90^{\circ}, 0)

  • (180∘,−1)(180^{\circ}, -1)

  • (270∘,0)(270^{\circ}, 0)

  • (360∘,1)(360^{\circ}, 1)

These points represent the maximum, intercepts, and minimum values of the cosine function. Understanding these key points and the periodic nature of the function is crucial for graphing transformations of the cosine function.

Transformations of Cosine Functions

The function $g(x) = \cos(x - 45^{\circ})$ is a transformation of the basic cosine function. Specifically, it represents a horizontal shift of the basic cosine function. The general form of a transformed cosine function is:

y=Acos⁡(B(x−C))+Dy = A \cos(B(x - C)) + D

Where:

  • A$ is the amplitude, which affects the vertical stretch or compression of the graph.

  • B$ affects the period of the function. The period is given by $\frac{360^{\circ}}{|B|}$.

  • C$ is the horizontal shift (phase shift). A positive value of $C$ shifts the graph to the right, and a negative value shifts it to the left.

  • D$ is the vertical shift. A positive value of $D$ shifts the graph upwards, and a negative value shifts it downwards.

In our case, $g(x) = \cos(x - 45^{\circ})$, we have $A = 1$, $B = 1$, $C = 45^{\circ}$, and $D = 0$. This means the graph of $g(x)$ is the same as the graph of $f(x) = \cos(x)$ but shifted $45^{\circ}$ to the right. Understanding these transformations is essential for accurately sketching the graph of $g(x)$. The horizontal shift of $45^{\circ}$ is the key to understanding the difference between the graphs of $f(x)$ and $g(x)$.

Sketching the Graph of g(x) = cos(x - 45°)

To sketch the graph of $g(x) = \cos(x - 45^{\circ})$, we'll follow these steps:

  1. Identify the key points of the basic cosine function: As mentioned earlier, the key points of $f(x) = \cos(x)$ are $(0^{\circ}, 1)$, $(90^{\circ}, 0)$, $(180^{\circ}, -1)$, $(270^{\circ}, 0)$, and $(360^{\circ}, 1)$.
  2. Apply the horizontal shift: Since $g(x) = \cos(x - 45^{\circ})$, we shift each of these key points $45^{\circ}$ to the right. This means we add $45^{\circ}$ to the x-coordinate of each point.
  3. Calculate the new key points: The new key points for $g(x)$ are:

    • (0∘+45∘,1)=(45∘,1)(0^{\circ} + 45^{\circ}, 1) = (45^{\circ}, 1)

    • (90∘+45∘,0)=(135∘,0)(90^{\circ} + 45^{\circ}, 0) = (135^{\circ}, 0)

    • (180∘+45∘,−1)=(225∘,−1)(180^{\circ} + 45^{\circ}, -1) = (225^{\circ}, -1)

    • (270∘+45∘,0)=(315∘,0)(270^{\circ} + 45^{\circ}, 0) = (315^{\circ}, 0)

    • (360∘+45∘,1)=(405∘,1)(360^{\circ} + 45^{\circ}, 1) = (405^{\circ}, 1)

  4. Plot the points: Plot these key points on a coordinate plane. Remember that we are considering the interval $x \in [0^{\circ}, 360^{\circ}]$, so we only need to plot the points within this range. We have the points $(45^{\circ}, 1)$, $(135^{\circ}, 0)$, $(225^{\circ}, -1)$, and $(315^{\circ}, 0)$. The point $(405^{\circ}, 1)$ is outside our interval, but it helps us visualize the periodic nature of the function.
  5. Connect the points: Draw a smooth curve through the plotted points, following the general shape of the cosine function. The graph should oscillate between 1 and -1, reaching its maximum at $(45^{\circ}, 1)$, its minimum at $(225^{\circ}, -1)$, and crossing the x-axis at $(135^{\circ}, 0)$ and $(315^{\circ}, 0)$.

By following these steps, you can accurately sketch the graph of $g(x) = \cos(x - 45^{\circ})$ within the specified interval. Accurate sketching relies on a clear understanding of the transformations and the key points of the function.

Identifying Intercepts and Turning Points

Once the graph is sketched, we can identify the intercepts and turning points. Intercepts are the points where the graph crosses the x-axis (x-intercepts) or the y-axis (y-intercept). Turning points are the points where the function changes direction, either from increasing to decreasing (maximum) or from decreasing to increasing (minimum).

Intercepts

  • X-intercepts: These are the points where $g(x) = 0$. From our graph and the calculated key points, we can see that the x-intercepts are at $(135^{\circ}, 0)$ and $(315^{\circ}, 0)$. These are the points where the cosine function equals zero.
  • Y-intercept: This is the point where $x = 0$. To find the y-intercept, we evaluate $g(0) = \cos(0 - 45^{\circ}) = \cos(-45^{\circ})$. Since cosine is an even function, $\cos(-45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \approx 0.707$. So, the y-intercept is approximately $(0^{\circ}, 0.707)$.

Turning Points

  • Maximum: This is the point where the function reaches its highest value. From the graph, we can see that the maximum occurs at $(45^{\circ}, 1)$. This is the peak of the cosine wave.
  • Minimum: This is the point where the function reaches its lowest value. From the graph, we can see that the minimum occurs at $(225^{\circ}, -1)$. This is the trough of the cosine wave.

Identifying these key features is crucial for a complete understanding of the function's behavior within the given interval. The intercepts tell us where the function crosses the axes, and the turning points tell us where the function changes direction.

Determining Values of x in the Interval [0°, 360°] for Specific Conditions

Finally, let's consider how to determine the values of $x$ in the interval $x \in [0^{\circ}, 360^{\circ}]$ that satisfy specific conditions. For example, we might want to find the values of $x$ where $g(x) = \frac{1}{2}$. To do this, we can use the graph or the unit circle, or we can solve the equation algebraically.

Graphical Method

  1. Draw a horizontal line: On the same graph as $g(x)$, draw a horizontal line at $y = \frac{1}{2}$.
  2. Find the intersection points: The points where the line intersects the graph of $g(x)$ represent the solutions to the equation $g(x) = \frac{1}{2}$.
  3. Read the x-coordinates: The x-coordinates of these intersection points are the values of $x$ that satisfy the condition.

From the graph, we can approximate the solutions, but for more accurate values, we can use the algebraic method.

Algebraic Method

  1. Set up the equation: We want to solve $\cos(x - 45^{\circ}) = \frac{1}{2}$.
  2. Find the reference angle: The reference angle is the angle whose cosine is $\frac{1}{2}$. We know that $\cos(60^{\circ}) = \frac{1}{2}$, so the reference angle is $60^{\circ}$.
  3. Find the general solutions: Since cosine is positive in the first and fourth quadrants, we have two general solutions:

    • x - 45^{\circ} = 60^{\circ} + 360^{\circ}k$, where $k$ is an integer.

    • x - 45^{\circ} = -60^{\circ} + 360^{\circ}k$, where $k$ is an integer.

  4. Solve for x:

    • x=105∘+360∘kx = 105^{\circ} + 360^{\circ}k

    • x=−15∘+360∘kx = -15^{\circ} + 360^{\circ}k

  5. Find solutions in the interval [0°, 360°]:
    • For $x = 105^{\circ} + 360^{\circ}k$, when $k = 0$, $x = 105^{\circ}$.
    • For $x = -15^{\circ} + 360^{\circ}k$, when $k = 1$, $x = 345^{\circ}$.

So, the values of $x$ in the interval $[0^{\circ}, 360^{\circ}]$ where $g(x) = \frac{1}{2}$ are $105^{\circ}$ and $345^{\circ}$. Solving trigonometric equations is a fundamental skill in mathematics, and understanding both graphical and algebraic methods is crucial.

Conclusion

In conclusion, sketching the graph of $g(x) = \cos(x - 45^{\circ})$ involves understanding the transformations of the basic cosine function, identifying key points, and accurately plotting the graph. Determining intercepts and turning points provides further insight into the function's behavior. Additionally, solving trigonometric equations allows us to find specific values of $x$ that satisfy certain conditions. By mastering these concepts, you can confidently analyze and graph cosine functions and their transformations. Mastering these concepts is essential for success in trigonometry and calculus. This comprehensive guide has provided you with the tools and knowledge to confidently tackle such problems. Remember to practice and apply these techniques to various examples to solidify your understanding.

Graphing cosine functions, Trigonometric functions, Transformations, Intercepts, Turning points, Horizontal shift, Phase shift, Cosine function, Amplitude, Period, Solving trigonometric equations, Key points, Periodic function, Trigonometry, Mathematics