8.2 Transformations of Logarithmic Functions

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Presentation transcript:

8.2 Transformations of Logarithmic Functions Logarithmic form: y = logb x Exponential Form: by = x. Exponent Exponent Base Base -2 -1 6 2 3 4 5 f (x) = 2x f (x) = log2 x y = x Math 30-1

Key Points from basic graph to transform include: Logarithmic Transformations Parameter Description of Transformation Mapping Notation a Vertical stretch by a factor of |a| (x, y) → (x, ay) b Horizontal stretch by a factor of (x, y) → h Horizontal translation right, h > 0 left, h < 0 (x, y) → (x + h, y) k Vertical translation up, k > 0 down, k < 0 (x, y) → (x, y + k) Key Points from basic graph to transform include: x-intercept (1, 0) Reflections??? vertical asymptote x = 0 Math 30-1

Sketching Graph of Logarithms Vertical asymptote x = 0 Vertical asymptote x = 0 Vertical asymptote x = 2 x-intercept at 1 x-intercept at 1 x-intercept at 3 Math 30-1

Determine the value of the missing coordinate. The point (32, b) is on the graph of b = 5 The point (a, 3) is on the graph of a = 27 The point (a, 4) is on the graph of a = 15 The point (19, b) is on the graph of b = -3 Math 30-1

Describe the transformations on the graph of y = log2x to become Horizontal stretch by a factor of 1/5 Horizontal translation of 2 units left Vertical translation of 3 units up The Vertical Asymptote , x = 0 translates to The domain of the image graph is Math 30-1

The graph of y = log x is transformed into the graph of y + 3 = log(x - 6) by a translation of 6 units __i__ and 3 units _ii_. The statement above is completed by the information in row Row i ii A right up B left C down D For the graph of , where 0 < b < 1, the domain is Math 30-1

The red graph can be generated by stretching the blue graph of y = log4 x. Write the equation that describes the red graph. y = log4 4x The red graph can be generated by stretching and reflecting the graph of y = log4 x. Write the equation that describes the red graph. y = -3log4 x Math 30-1

The solid graph can be generated by translating the dashed graph of y = log4 x. Write the equation that describes the solid graph. y = log4 (x) + 1 The graph of y = log2 x has been vertically stretched about the x-axis by a factor of 3 , horizontally stretched about the y-axis by a factor of 1/5 , reflected in the x-axis, and translated of 7 units left and 2 units up. Write the equation of the transformed function in the form y = a log2 (b(x - h) + k. y = -3 log2 (5(x + 7) + 2 Math 30-1

Assignment Page 389 1, 2, 4a,c, 5b,d, 6a,c, 7, 8b, 9, 10, 13, 14, 16a Math 30-1