Transformations of Exponential, Logarithmic, and Trigonometric Functions

Any function y = f(x) can be transformed into the form y = af(k(x-d)) + c. Each parameter affects the graph in a specific way:\n- 'a': Vertical stretch or compression by a factor of |a|. If a < 0, there is a reflection across the x-axis.\n- 'k': Horizontal stretch or compression by a factor of 1/|k|. If k < 0, there is a reflection across the y-axis.\n- 'd': Horizontal translation (phase shift). The graph shifts 'd' units to the right if d > 0, and 'd' units to the left if d < 0.\n- 'c': Vertical translation. The graph shifts 'c' units up if c > 0, and 'c' units down if c < 0.\n\nThe order of applying transformations is crucial: stretches, compressions, and reflections are applied first, followed by translations.

An exponential function in its base form is y = b^x, where b > 0 and b ≠ 1. The transformed form is y = a * b^(k(x-d)) + c.\n- The horizontal asymptote of the base function y = b^x is y = 0. For the transformed function, the horizontal asymptote is y = c.\n- The domain is always x ∈ R.\n- The range is y > c if a > 0, and y < c if a < 0.

A logarithmic function in its base form is y = log_b(x), where b > 0 and b ≠ 1. This is the inverse of y = b^x. The transformed form is y = a * log_b(k(x-d)) + c.\n- The vertical asymptote of the base function y = log_b(x) is x = 0. For the transformed function, the vertical asymptote is x = d.\n- The domain is x > d if k > 0, and x < d if k < 0.\n- The range is always y ∈ R.

Logarithms follow specific rules that are essential for simplifying expressions and solving equations:\n- Product Rule: The logarithm of a product is the sum of the logarithms.\n- Quotient Rule: The logarithm of a quotient is the difference of the logarithms.\n- Power Rule: The logarithm of a power is the exponent times the logarithm of the base.\n- Change of Base Formula: Allows conversion between different logarithm bases.

The base forms are y = sin(x) and y = cos(x). The transformed form is y = a * sin(k(x-d)) + c or y = a * cos(k(x-d)) + c.\n- 'a': Amplitude = |a|. This is half the distance between the maximum and minimum values. If a < 0, there is a reflection across the x-axis.\n- 'k': Determines the period. Period = (2π / |k|) for radians or (360° / |k|) for degrees. If k < 0, there is a reflection across the y-axis.\n- 'd': Phase shift. Horizontal translation. 'd' units right if d > 0, 'd' units left if d < 0.\n- 'c': Vertical shift. The equation of the axis of the curve is y = c.\n- Maximum value = c + |a|, Minimum value = c - |a|.