Math Study Notes: Functions, Transformations, and Graphs

Question 1 Describe the transformations and sketch the graphs of the following trigonometric functions: $ a(x) = -2 \cos(2x) + 3 $, $ b(x) = 6 $, $ c(x) = 3 \sin(x - \pi) - 1 $, $ d(x) = \tan(x) $. Answer: Step 1: $ a(x) = -2 \cos(2x) + 3 $ We know $ y = A \sin(Bx - C) + D $ is the general equation of a sin function, where $ |A| $ is amplitude, $\frac{2\pi}{B}$ is time period, $ \frac{C}{B} $ is phase shift, and $ D $ is vertical shift. $ f(x) = -2 \cos(2x) + 3 $ Amplitude = $ 2 $ Time Period = $ \frac{2\pi}{2} = \pi $ Phase shift = $ \frac{0}{2} = 0 $ Vertical shift = 3 [Graph of $ a(x) = -2 \cos(2x) + 3 $] Step 2: $ c(x) = 3 \sin(x - \pi) - 1 $ Amplitude = 3 Time Period = $ \frac{2\pi}{1} = 2\pi $ Phase shift = $ \pi $ Vertical shift = -1 [Graph of $ c(x) = 3 \sin(x - \pi) - 1 $] Note: $ b(x) = 6 $ is a constant function (horizontal line at y = 6), and $ d(x) = \tan(x) $ is the tangent function with a period of $\pi$ and vertical asymptotes at $ x = \frac{\pi}{2} + k\pi $ (where k is an integer). Sketches would reflect these properties accordingly. Question 2 Begin by graphing $ f(x) = \log_2 x $. Then use transformations of this graph to graph the given function. What is the graph? Answer: Step 1: Graph of $ f(x) = \log_2 x $ [Graph of $ f(x) = \log_2 x $ with vertical asymptote at $ x = 0 $] Step 2: Compare given equation with general equation of a sin function, i.e., $ y = A \sin(Bx - C) + D $ The problem seems to imply a transformation, but no specific function is provided beyond the initial setup. Assuming a typo or incomplete context, if the intent was to graph a transformed logarithmic function (e.g., based on prior context), it would involve shifts or reflections. However, based on the image, no additional function is specified beyond $ f(x) = \log_2 x $. Question 3 In your earlier work in algebra, you learned how to recognize linear, exponential, and quadratic functions by the form of their symbolic rules. Geometric transformations also can be recognized by their symbolic rules. What transformation is defined by each of the following? $ a. (x, y) \rightarrow (5x, 5y) $, $ b. (x, y) \rightarrow \left(\frac{1}{2} - x, -y\right) $, $ c. (x, y) \rightarrow (4x - 12, 4y + 8) $ Answer: Step 1: $ a. (x, y) \rightarrow (5x, 5y) $ This symbolic rule corresponds to a composite transformation of two transformations. Given in the order in which they should be applied: Size transformation of magnitude 5 around the origin Rotation of 180° around the origin The symbolic rule is given to us as $ (x, y) \rightarrow (5x, 5y) $ Step 2: $ b. (x, y) \rightarrow \left(\frac{1}{2} - x, -y\right) $ This symbolic rule corresponds to a composite transformation of two transformations. Given in the order in which they should be applied: Reflection across the line $ y = x $ Translation of horizontal component -1/2 and vertical component 0 The symbolic rule is given to us as $ (x, y) \rightarrow \left(\frac{1}{2} - x, -y\right) $ Step 3: $ c. (x, y) \rightarrow (4x - 12, 4y + 8) $ This symbolic rule corresponds to a composite transformation of two transformations. Given in the order in which they should be applied: Size transformation of magnitude 4 around the origin Translation of horizontal component -3 and vertical component 2 The symbolic rule is given to us as $ (x, y) \rightarrow (4x - 12, 4y + 8) $ Question 4 Transform the equation $ r = 12 $ from polar to rectangular form. Answer: Given: $ r = 12 $ Square to get $ r^2 $ $ r^2 = 12^2 $ Conversion formula: $ x^2 + y^2 = r^2 $ Substitute: $ x^2 + y^2 = 144 $ Equation 5 Question 5 The line $ y = \frac{4}{5} x + 2 $ passes through (1, 2). Express a line parallel to it with an equation. Answer: The lines are parallel when their $ m $ (slope) is the same. $ m = \frac{4}{5} $ Find $ b $ by substituting $ m = \frac{4}{5} $ and (1, 2): $ 2 = \frac{4}{5}(1) + b $ $ 2 = \frac{4}{5} + b $ $ b = 2 - \frac{4}{5} $ $ b = \frac{10}{5} - \frac{4}{5} $ $ b = \frac{6}{5} $ Thus, the expression is: $ y = \frac{4}{5} x + \frac{6}{5} $