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Trig 4.2 Translations of the Graphs of Sine & Cosine Functions

Amy Arel・2 minutes read

The text outlines the equations and principles for graphing sine and cosine functions, detailing the roles of amplitude, period, vertical translation, and phase shift in shaping the graphs. It emphasizes the importance of practicing these transformations and understanding their effects on the graph's characteristics, preparing students for more complex functions in future lessons.

Insights

  • The equations for sine and cosine graphs, \( y = c + a \sin(b(x - d)) \) and \( y = c + a \cos(b(x - d)) \), define key parameters: amplitude \( a \) affects height, period \( \frac{2\pi}{b} \) determines width, vertical translation \( c \) shifts the graph up or down, and phase shift \( d \) moves it left or right, emphasizing the need for careful calculation of these values.
  • The vertical translation \( c \) modifies the midline of the graph, which may no longer align with \( y = 0 \), while the phase shift \( d \) is crucial for determining the starting position of the graph, as a positive \( d \) indicates a rightward shift and a negative \( d \) indicates a leftward shift.
  • Understanding the "jump" between key points on the graph, calculated as \( \frac{\text{period}}{4} \), is essential for accurately plotting the sine and cosine functions, with the graph beginning at the phase shift \( d \) and subsequent points derived from adding the calculated jump.
  • The text highlights the importance of practicing graphing techniques and recognizing variations in problems, such as different signs in the equations, to prepare students for future topics in calculus related to transformations and trigonometric functions.
  • Each parameter's effect on the graph is summarized: amplitude \( a \) influences height, period \( b \) affects width, vertical translation \( c \) shifts the graph, and phase shift \( d \) determines the starting point, underscoring the need for a solid grasp of these concepts for graphing sine and cosine functions effectively.

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Recent questions

  • What is the definition of amplitude?

    Amplitude is the maximum distance from the midline.

  • How do I graph a sine function?

    To graph a sine function, identify amplitude, period, phase shift, and vertical translation.

  • What is a phase shift in graphs?

    A phase shift is a horizontal translation of the graph.

  • What does vertical translation mean?

    Vertical translation shifts the graph up or down.

  • How is the period of a function calculated?

    The period is calculated as \( \frac{2\pi}{b} \).

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Summary

00:00

Graphing Sine and Cosine Functions Explained

  • The equations for translating sine and cosine graphs are given as \( y = c + a \sin(b(x - d)) \) and \( y = c + a \cos(b(x - d)) \), where \( a \) represents amplitude, \( b \) indicates horizontal stretch or shrink, \( c \) is the vertical translation, and \( d \) is the phase shift.
  • Amplitude \( a \) determines the vertical stretch or shrink of the graph, while the period is calculated as \( \frac{2\pi}{b} \). It is crucial that \( b \) is in factored form, such as \( b(x + 3) \), to ensure accurate calculations.
  • The vertical translation \( c \) shifts the graph up if positive and down if negative, affecting the midline of the graph, which may no longer be at \( y = 0 \).
  • The phase shift \( d \) translates the graph horizontally; a positive \( d \) moves the graph to the right, while a negative \( d \) moves it to the left. The sign of \( d \) is critical, as it is derived from the equation's structure.
  • The jump, or the distance between key points on the graph, is calculated as \( \frac{\text{period}}{4} \), and the graph starts at the phase shift \( d \), with subsequent points determined by adding the jump.
  • To graph, first find the amplitude to establish maximum and minimum values, and consider using dashed lines to indicate these limits for clarity.
  • The steps for graphing include: determining amplitude, calculating the period as \( \frac{2\pi}{b} \), identifying vertical translation \( c \), finding phase shift \( d \), and calculating the jump.
  • An example provided involves graphing \( y = \sin(x - \frac{\pi}{3}) \) over one period, where the amplitude \( a = 1 \), period \( = 2\pi \), and phase shift \( d = \frac{\pi}{3} \), indicating a rightward shift.
  • The starting point for the graph is at \( \frac{\pi}{3} \), and the ending point is calculated by adding the period \( 2\pi \) to the starting point, resulting in \( \frac{7\pi}{3} \).
  • The graphing process involves marking key points based on the calculated jumps, ensuring that the graph reflects the sine function's behavior, starting at the phase shift and following the established amplitude and period.

15:52

Understanding Transformations in Sine Functions

  • The text discusses the importance of understanding transformations in calculus, particularly through the use of tables and graphs, emphasizing a personal struggle with the table method over 40 years ago.
  • It introduces the concept of calculating angles for sine functions, demonstrating how to find true angles by subtracting fractions, such as \( \frac{5\pi}{6} - \frac{2\pi}{6} = \frac{3\pi}{6} \), which simplifies to \( \frac{\pi}{2} \).
  • The sine values for key angles are provided: \( \sin(0) = 0 \), \( \sin\left(\frac{\pi}{2}\right) = 1 \), \( \sin(\pi) = 0 \), \( \sin\left(\frac{3\pi}{2}\right) = -1 \), and \( \sin(2\pi) = 0 \), which can be plotted on a graph.
  • The text explains the difference between horizontal and vertical translations in sine functions, noting that parentheses around the x-value indicate horizontal shifts, while those affecting the y-value indicate vertical shifts.
  • An example is given for graphing \( y = 3 \cos\left(x + \frac{\pi}{4}\right) \) over one period, identifying the amplitude \( a = 3 \) and the period \( 2\pi \), with a horizontal shift of \( d = -\frac{\pi}{4} \).
  • The graphing process involves marking key points based on the period and amplitude, starting at \( -\frac{\pi}{4} \) and making jumps of \( \frac{2\pi}{4} \) to find subsequent points.
  • The text emphasizes the importance of practicing graphing techniques, noting that students should be prepared for variations in problems, such as different signs in the equations.
  • A new example is introduced for graphing \( y = -2 \cos(3x + \pi) \) over two periods, requiring factoring out the 3 to find the correct d-value and period, which is \( \frac{2\pi}{3} \).
  • The a-value is identified as \( -2 \), indicating a reflection across the x-axis, while the d-value is calculated as \( \frac{\pi}{3} \), which determines the starting point of the graph.
  • The text concludes with a reminder to check calculations for accuracy, particularly regarding signs and values, to ensure correct graphing and understanding of transformations.

31:43

Graphing Cosine Functions with Amplitude and Period

  • The process begins by determining the "jump" for the function, calculated by taking the period \( \frac{2\pi}{3} \) and dividing it by 4, resulting in \( \frac{2\pi}{12} \), which simplifies to \( \frac{\pi}{6} \). This value is used for subsequent calculations.
  • The next step involves converting \( \frac{2\pi}{6} \) to a common denominator of 6 for easier calculations, resulting in \( -\frac{2\pi}{6} \) as the starting point, which is then incremented by \( \frac{\pi}{6} \) for each step.
  • The increments are calculated as follows: starting from \( -\frac{2\pi}{6} \), adding \( \frac{\pi}{6} \) results in \( -\frac{\pi}{6} \), \( 0 \), \( \frac{\pi}{6} \), and \( \frac{2\pi}{6} \), marking four key points.
  • The problem specifies to graph over two periods, allowing the choice to move either to the right or left. Moving to the right involves adding \( \frac{\pi}{6} \) repeatedly, while moving left requires subtracting \( \frac{\pi}{6} \).
  • For the leftward movement, starting from \( -\frac{2\pi}{6} \), the values are calculated as \( -\frac{3\pi}{6} \), \( -\frac{4\pi}{6} \), \( -\frac{5\pi}{6} \), and \( -\frac{6\pi}{6} \), marking the second period.
  • The graphing process involves marking tick marks for each calculated point, ensuring to label the midline and the maximum and minimum values, which are determined by the amplitude of 2, reflecting the cosine function.
  • The amplitude is defined as the absolute value of the coefficient in front of the cosine function, which is 2, indicating the graph will oscillate between 2 and -2.
  • The vertical translation is determined by the constant \( c \) in the function \( y = 3 - 2\cos(3x) \), which shifts the midline up by 3 units, affecting the overall positioning of the graph.
  • The method for graphing involves identifying the period \( \frac{2\pi}{3} \), dividing it into four equal parts, and evaluating the function at these points to find maximum, minimum, and midline intersection points.
  • Two methods for graphing are presented: one using a table of values and the other focusing on amplitude and translations, with the latter being preferred for its straightforward approach to visualizing the function's behavior.

48:57

Understanding Sine Graph Transformations and Parameters

  • The equation given is \( y = -1 + 2 \sin(4x + \frac{\pi}{2}) \), which includes parameters \( a \), \( b \), \( c \), and \( d \) that need to be identified and factored for graphing purposes. The first step is to rewrite the equation in the form \( y = c + a \sin(b(x - d)) \).
  • The values identified are \( a = 2 \), \( b = 4 \), \( c = -1 \), and \( d = \frac{\pi}{4} \). The amplitude \( a \) is positive, indicating no reflection across the x-axis, and the period is calculated as \( \frac{2\pi}{b} = \frac{2\pi}{4} = \frac{\pi}{2} \).
  • The midline of the graph is adjusted down by 1 unit due to the \( c \) value, resulting in a new midline at \( y = -1 \). The maximum value is \( -1 + 2 = 1 \) and the minimum value is \( -1 - 2 = -3 \).
  • The phase shift \( d \) is \( \frac{\pi}{4} \), indicating the graph starts at \( \frac{\pi}{4} \) to the left. The scaling for the graph is determined by dividing the period \( \frac{\pi}{2} \) into four segments, resulting in a jump of \( \frac{\pi}{8} \).
  • The starting point for the graph is adjusted to \( -\frac{2\pi}{8} \) or \( -\frac{\pi}{4} \), and the graph is plotted by adding \( \frac{\pi}{8} \) for each subsequent point, resulting in points at \( -\frac{\pi}{4}, -\frac{\pi}{8}, 0, \frac{\pi}{8}, \frac{\pi}{4} \).
  • The graph is sinusoidal, starting at the midline, moving up to the maximum, back to the midline, down to the minimum, and returning to the midline, confirming the periodic nature of the sine function.
  • The effects of parameters on the graph are summarized: \( a \) determines amplitude, \( b \) affects the period, \( c \) indicates vertical translation, and \( d \) represents horizontal translation. A negative \( a \) would reflect the graph across the x-axis.
  • An example homework problem is presented, \( y = -3 + 5 \sin(x + \frac{\pi}{6}) \), where the amplitude is 5, the period is \( 2\pi \), the vertical shift is down 3, and the phase shift is \( -\frac{\pi}{6} \) (left).
  • The process for inputting values into a graphing tool is described, emphasizing the need to identify whether the graph is sine or cosine, and to input the amplitude, period, vertical shift, and phase shift correctly.
  • The importance of understanding the effects of each parameter on the graph is highlighted, as well as the necessity of showing work on paper for grading purposes, ensuring comprehension of the transformations involved in graphing sine and cosine functions.

01:04:37

Graph Transformations and Midline Shifts Explained

  • When moving down 3 units on a graph, the midline shifts accordingly, resulting in a new maximum and minimum; for example, if the original maximum was at 5, it will now be at 2, and if the original minimum was at 0, it will now be at -3. Additionally, the point originally at (0, 0) will move to (-π/6, -3), illustrating the graph's transformation. Understanding how to graph these transformations by hand is essential, as future lessons will cover tangent, cotangent, secant, and cosecant functions, requiring a solid grasp of sections 4.1 and 4.2.
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