Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

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Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problema 25

Capitolo 2, Problema 25

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2x + π/2)

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Identify the general form of the sine function: $y = A \sin(Bx + C)$, where $A$ is the amplitude, $B$ affects the period, and $C$ affects the phase shift.

Find the amplitude by taking the absolute value of the coefficient in front of the sine function: $\text{Amplitude} = |A| = |-2|$.

Calculate the period using the formula $\text{Period} = \frac{2\pi}{|B|}$, where $B$ is the coefficient of $x$ inside the sine function. Here, $B = 2$.

Determine the phase shift by solving $Bx + C = 0$ for $x$, which gives $x = -\frac{C}{B}$. Substitute $C = \frac{\pi}{2}$ and $B = 2$ to find the phase shift.

To graph one period of the function, start at the phase shift on the x-axis, plot key points at intervals of $\frac{\text{Period}}{4}$, and use the amplitude to mark the maximum and minimum values of the sine wave.

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Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of the function's output, representing the height from the midline to the peak. For sine and cosine functions, it is the absolute value of the coefficient before the sine or cosine term. In y = −2 sin(2x + π/2), the amplitude is |−2| = 2.

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Introduction to Trigonometric Functions

Period of a Trigonometric Function

The period is the length of one complete cycle of the function, calculated as 2π divided by the coefficient of x inside the function. For y = −2 sin(2x + π/2), the coefficient is 2, so the period is 2π/2 = π. This means the function repeats every π units.

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Period of Sine and Cosine Functions

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving the inside of the function's argument for zero. For y = −2 sin(2x + π/2), set 2x + π/2 = 0, giving x = −π/4. This means the graph shifts left by π/4 units.

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Phase Shifts

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In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

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In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 sin(x + π/2)

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Domanda del libro di testo

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 1/2 sin(x + π)