OnRamps PreCalculus Unit 4B Quiz Review

Key features of the Parent Sine Function f(x) = \sin(x):

1. Domain: (-\infty, \infty)

2. Range: [-1, 1]

3. Period: 2\pi

4. Amplitude: 1

5. Key Points: (0,0), (\frac{\pi}{2}, 1), (\pi, 0), (\frac{3\pi}{2}, -1), (2\pi, 0)

Key features of the Parent Cosine Function f(x) = \cos(x):

1. Domain: (-\infty, \infty)

2. Range: [-1, 1]

3. Period: 2\pi

4. Amplitude: 1

5. Key Points: (0, 1), (\frac{\pi}{2}, 0), (\pi, -1), (\frac{3\pi}{2}, 0), (2\pi, 1)

Parent Tangent Function f(x) = \tan(x) characteristics:

• Period: \pi

• Vertical Asymptotes: x = \frac{\pi}{2} + n\pi for any integer n

• x-intercepts: (n\pi, 0)

• Range: (-\infty, \infty)

Horizontal Transformations of Tangent f(x) = \tan(bx):

• New Period: \frac{\pi}{|b|}

• Horizontal Stretch: Occurs when 0 < |b| < 1

• Horizontal Compression: Occurs when |b| > 1

• Asymptote Location: Set the argument bx = \pm \frac{\pi}{2} and solve for x

General Form of a Sinusoidal Function:

y = a \sin(b(x - h)) + k or y = a \cos(b(x - h)) + k

• |a|: Amplitude

• b: Frequency coefficient (\text{Period} = \frac{2\pi}{b}, for b > 0)

• h: Phase shift (horizontal shift)

• k: Midline (vertical shift)

Sinusoidal Modeling: Calculating Amplitude (a) and Midline (k):

• Amplitude (a): \frac{\text{Maximum} - \text{Minimum}}{2}

• Midline (k): \frac{\text{Maximum} + \text{Minimum}}{2}

• Note: If modeling a periodic wave, the vertical distance from midline to peak is the amplitude.

Inverse Sine Function (f(x) = \sin^{-1}(x) or \arcsin(x)):

• Domain: [-1, 1]

• Range (Principal Values): [-\frac{\pi}{2}, \frac{\pi}{2}]

• Purpose: Returns the angle \theta in the restricted interval whose sine is x

Inverse Cosine Function (f(x) = \cos^{-1}(x) or \arccos(x)):

• Domain: [-1, 1]

• Range (Principal Values): [0, \pi]

• Purpose: Returns the angle \theta in the restricted interval whose cosine is x

Inverse Tangent Function (f(x) = \tan^{-1}(x) or \arctan(x)):

• Domain: (-\infty, \infty)

• Range (Principal Values): (-\frac{\pi}{2}, \frac{\pi}{2})

• Note: The endpoints of the range are not included because the tangent function has asymptotes at those values