Domanda del libro di testo
In Exercises 63–82, use a sketch to find the exact value of each expression. cot (csc⁻¹ 8)
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 77
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 77Capitolo 2, Problema 77
In Exercises 75–78, graph one period of each function. y = −|3 sin πx|
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Identify the base trigonometric function inside the absolute value, which is \(\sin(\pi x)\), and note that its period is given by \(\frac{2\pi}{B}\) where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).
Since the function is \(y = -|3 \sin(\pi x)|\), first consider the inner function \(3 \sin(\pi x)\). The amplitude of \(\sin(\pi x)\) is 1, so multiplying by 3 scales the amplitude to 3.
The absolute value \(|3 \sin(\pi x)|\) means all negative values of \(3 \sin(\pi x)\) become positive, reflecting any part of the graph below the x-axis to above the x-axis.
The negative sign outside the absolute value, \(- |3 \sin(\pi x)|\), flips the entire graph of \(|3 \sin(\pi x)|\) over the x-axis, making all values non-positive (zero or negative).
To graph one period, plot the function from \(x=0\) to \(x=2\), marking key points such as where \(\sin(\pi x)\) is zero, positive, or negative, apply the absolute value, then apply the negative sign, and connect these points smoothly to reflect the shape of the transformed sine wave.
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Period of a Sine Function
The period of a sine function is the length of one complete cycle on the x-axis. For y = sin(bx), the period is given by 2π / b. Understanding the period helps in determining the interval over which to graph one full cycle of the function.
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Percorso guidato
5:33
Period of Sine and Cosine Functions
Absolute Value Transformation
Applying the absolute value to a function, such as |f(x)|, reflects all negative values of f(x) above the x-axis, making the output non-negative. This changes the shape of the graph by 'folding' any parts below the x-axis upwards.
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Percorso guidato
5:25Introduction to Transformations
Vertical Scaling and Reflection
Multiplying a function by a constant scales it vertically, stretching or compressing the graph. A negative multiplier, like -|3 sin πx|, reflects the graph across the x-axis, flipping it upside down, which affects the amplitude and orientation of the graph.
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Percorso guidato
5:00Reflections of Functions
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In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = 2 cos x, g(x) = cos 2x, h(x) = (f + g)(x)
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In Exercises 75–78, graph one period of each function. y = |2 cos x/2|
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