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How do you find the period of \(y = 3\tan(2x + \pi) - 1\)?
1. Rewrite as \(y = 3\tan(2(x + \frac{\pi}{2})) - 1\). 2. Identify \(b = 2\). 3. Use the formula \(T = \frac{\pi}{|b|} = \frac{\pi}{2}\).
How do you determine the phase shift of \(y = \tan(x - \frac{\pi}{4})\)?
1. Identify 'c' in the general equation \(y = a \tan(b(x - c)) + d\). 2. In this case, \(c = \frac{\pi}{4}\). 3. The phase shift is \(\frac{\pi}{4}\) to the right.
How do you find the vertical asymptotes of \(y = \tan(2x)\) within the interval \([0, \pi]\)?
1. Set \(2x = \frac{\pi}{2} + k\pi\). 2. Solve for \(x\): \(x = \frac{\pi}{4} + \frac{k\pi}{2}\). 3. Find values of k that place x in \([0, \pi]\). For k=0, \(x = \frac{\pi}{4}\). For k=1, \(x = \frac{3\pi}{4}\).
How do you graph \(y = -\tan(x)\)?
1. Start with the graph of \(y = \tan(x)\). 2. Reflect the graph over the x-axis. This means every y-value becomes its opposite.
How do you determine the vertical shift of \(y = \tan(x) + 2\)?
1. Identify 'd' in the general equation \(y = a \tan(b(x - c)) + d\). 2. In this case, \(d = 2\). 3. The vertical shift is 2 units upward.
Given the graph of \(y = \tan(x)\), how do you sketch \(y = \tan(\frac{1}{2}x)\)?
1. Identify that \(b = \frac{1}{2}\), which means the period will be \(\frac{\pi}{1/2} = 2\pi\). 2. Stretch the graph horizontally by a factor of 2. The asymptotes will now be at \(x = \pi + k2\pi\).
How do you find the equation of a tangent function with a period of \(\frac{\pi}{3}\)?
1. Use the formula \(T = \frac{\pi}{|b|}\). 2. Set \(\frac{\pi}{3} = \frac{\pi}{|b|}\). 3. Solve for \(b\): \(b = 3\). 4. The equation is \(y = \tan(3x)\) (assuming no phase or vertical shift).
How do you find the domain of \(y = \tan(x)\)?
1. Identify where \(\cos(x) = 0\). 2. These are the points where the tangent function is undefined. 3. The domain is all real numbers except \(x = \frac{\pi}{2} + k\pi\), where k is an integer.
How do you find the range of \(y = a\tan(bx + c) + d\)?
1. The range of the tangent function is always all real numbers, unless there are restrictions given in the problem. 2. Therefore the range is \((-\infty, \infty)\).
How to solve \(\tan(x) = 1\) for \(x\)?
1. Recognize that \(\tan(\frac{\pi}{4}) = 1\). 2. Since the period of tangent is \(\pi\), the general solution is \(x = \frac{\pi}{4} + k\pi\), where k is an integer.
What are the key differences between the graphs of \(\tan(x)\) and \(\sin(x)\)?
Tangent: Vertical asymptotes, period of \(\pi\), range of \((-\infty, \infty)\). Sine: No asymptotes, period of \(2\pi\), range of \([-1, 1]\).
Compare the period of \(\tan(x)\) and \(\cos(x)\).
Tangent: Period is \(\pi\). Cosine: Period is \(2\pi\).
Compare the range of \(\tan(x)\) and \(\sin(x)\).
Tangent: Range is \((-\infty, \infty)\). Sine: Range is \([-1, 1]\).
What are the key differences between transformations of \(\tan(x)\) and \(\sin(x)\)?
Period changes are calculated differently. Tangent: \(T = \frac{\pi}{|b|}\). Sine: \(T = \frac{2\pi}{|b|}\). Asymptotes exist for tangent but not for sine.
Compare the asymptotes of \(\tan(x)\) and \(\cot(x)\).
Tangent: Asymptotes at \(x = \frac{\pi}{2} + k\pi\). Cotangent: Asymptotes at \(x = k\pi\) (where k is an integer).
Compare the behavior of \(\tan(x)\) and \(\sin(x)\) near \(x = 0\).
As \(x\) approaches 0, \(\tan(x)\) approaches 0. As \(x\) approaches 0, \(\sin(x)\) approaches 0.
Compare the symmetry of \(\tan(x)\) and \(\cos(x)\).
Tangent: Odd function, symmetric about the origin. Cosine: Even function, symmetric about the y-axis.
Compare the domain of \(\tan(x)\) and \(\sin(x)\).
Tangent: All real numbers except \(x = \frac{\pi}{2} + k\pi\). Sine: All real numbers.
Compare the graphs of \(y = \tan(x)\) and \(y = \cot(x)\).
Tangent: Increasing between asymptotes. Cotangent: Decreasing between asymptotes. Asymptotes are located in different places.
Compare the effect of 'a' in \(y = a\tan(x)\) and \(y = a\sin(x)\).
Tangent: Vertical stretch/compression, reflection if 'a' is negative. Sine: Amplitude change, reflection if 'a' is negative.
What is the formula for \(\tan(\theta)\) in terms of \(\sin(\theta)\) and \(\cos(\theta)\)?
\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
What is the general equation for a transformed tangent function?
\(y = a \tan(b(x - c)) + d\)
What is the formula for the period \(T\) of a tangent function?
\(T = \frac{\pi}{|b|}\)
How do you calculate the location of vertical asymptotes for \(y = \tan(x)\)?
\(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer.
How does 'a' affect the tangent function \(y = a \tan(x)\)?
'a' controls the vertical dilation. If 'a' is negative, the function is reflected over the x-axis.
How does 'b' affect the tangent function \(y = \tan(bx)\)?
'b' affects the period of the function. The period is \(T = \frac{\pi}{b}\).
How does 'c' affect the tangent function \(y = \tan(x-c)\)?
'c' shifts the graph horizontally. Positive 'c' shifts right, negative 'c' shifts left.
How does 'd' affect the tangent function \(y = \tan(x)+d\)?
'd' shifts the graph vertically. Positive 'd' shifts up, negative 'd' shifts down.
What is \(x\) on the unit circle?
\(x = cos(\theta)\)
What is \(y\) on the unit circle?
\(y = sin(\theta)\)