Exponent Rules, Logarithms, and Trigonometric Identities Flashcards

This flashcard set covers exponent properties, logarithmic transformations, conversion between polar and rectangular coordinates, and parameters for trigonometric function modeling.

Exponent Product Rule

$$a^x \times a^y = a^{x+y}$$

Logarithm Product Rule

$$\log_{10}(a \times b) = \log_{10}(a) + \log_{10}(b)$$

Logarithm Quotient Rule

$$\log_{10}(\frac{a}{b}) = \log_{10}(a) - \log_{10}(b)$$

Logarithm Power Rule

$$\log_{10}(a^b) = b(\log_{10}(a))$$

Exponential Form of a Logarithm

$$\log_b(x) = y \rightarrow b^y = x$$

Polar to Rectangular Conversion (x)

$$x = r(\cos(\theta))$$

Polar to Rectangular Conversion (y)

$$y = r(\sin(\theta))$$

Rectangular to Polar Conversion (r)

$$r = \sqrt{x^2 + y^2}$$

Rectangular to Polar Conversion (Angle)

$\theta = \tan^{-1}(\frac{y}{x})$ (then find $\theta$ on the unit circle)

Amplitude (A)

The variable $A$ in the function $f(x) = A\cos(B(x+C)) + D$

Frequency (B)

$\frac{2\pi}{P}$ for $\cos$ and $\sin$, or $\frac{\pi}{P}$ for $\tan$

Phase Shift (C)

The horizontal shift starting from the $0$ midline

Midline (D)

The vertical shift of the function, represented as the variable $D$

Concave Down

The curvature of a function characterized by a downward-opening shape

Concave Up

The curvature of a function characterized by an upward-opening shape