Review L11-15 Answer Key PDF

# Review (Lessons 11-15) ## Combining Transformations 1. **Consider the graph shown.** * A graph is shown with an origin and x and y axes ranging from -5 to 5. The curve goes through the points (-3, 4), (0, 4), (0, -3) and (3, 0). **a. Draw the image graph given the mapping notation (x, y) → (x...

# Review (Lessons 11-15) ## Combining Transformations 1. **Consider the graph shown.** * A graph is shown with an origin and x and y axes ranging from -5 to 5. The curve goes through the points (-3, 4), (0, 4), (0, -3) and (3, 0). **a. Draw the image graph given the mapping notation (x, y) → (x + 3, -y)** **b. Identify the new domain, range and intercepts. D = [1, 3] R = [-4, -2]** **c. Explain how the transformations affected the domain. Narrows the domain** 2. **A transformation of the graph of y = f(x) is represented by the equation y = -2f(x + 5) + 1. The point (7, 5) lies on the transformed graph. What are the coordinates of the corresponding point on the graph of y = f(x)? (4, -2)** 3. **The function y = 3|2x-8| is best described as y = |x| after:** * a. a vertical stretch of 1/3, a horizontal stretch of 2, and a shift of 8 units to the right. * b. a vertical stretch of 3, a horizontal stretch of 1/2, and a shift of 8 units to the right. * **c. a vertical stretch of 1/3, a horizontal stretch of 2, and a shift of 4 units to the right.** * **d. a vertical stretch of 3, a horizontal stretch of 1/2, and a shift of 4 units to the right.** ## Unit Circle and Trig Ratios: 1. Point A(√4/2, √1/2) and point B(-√1/2, -√3/2) lie on the terminal arm of two different angles in standard position. The angle, θ where 0 ≤ θ < π, can be expressed in the form aπ/b = π/12. * An image of a unit circle is shown with point A being in quadrant 1 and point B being in quadrant 3. 2. For the angles 5π/6, 7π/6, 11π/6, the following statements are given. * **Statement 1: They all have the same reference angle.** * Statement 2: These angles in degrees are, respectively, 30°, 150°, 210°, and 300°. * Statement 3: They are all part of the solution set θ = π/6 + 2ηπ, η∈I. * **Statement 4: The values of sin(5π/6) and sin(11π/6) are positive.** **Two statements that are true from the list are numbered 1 and 4.** 3. Given that csc θ = 5/8, where π/2 < θ < 3π/2, determine the exact value of cos θ. cos θ = -√39/8 4. If tan θ = -√3, where 0 ≤ θ ≤ 2π, then the largest positive value of θ, to the nearest tenth is 4.3 rad. 5. The point (k, -3/5) is on the unit circle and it is located in quadrant III. If the principal angle it forms is θ, then determine the exact value of cos θ. cos θ = -3/5 ## Composite Functions: 1. The graphs of y = f(x) and y = g(x) are shown. Use the graph to estimate the value of each expression to 1 decimal place. * A graph of y = f(x) and y = g(x) is shown for a range of x = -12 to +12 with a range of y = -12 to +12. **a. (f o g)(4) ≈ -4.2** **b. g(f(-7)) ≈ -0.5** **c. (f o f)(6) ≈ 4.2** **d. g(g(-10)) ≈ -5.2** **e. f(g(0)) - g(1) ≈ -2.7** **f. (g o f)(9) +f(-11) ≈ -7** 2. For each pair of functions below, determine (f o g)(x), (g o f)(x), (f o f)(x) and (g o g)(x). **a. f(x) = x^2 -1 and g(x) = x + 1** *(f o g)(x) = (x + 1)^2 -1 = x^2 + 2x* *(g o f)(x) = (x^2 -1) + 1 = x^2* *(f o f)(x) = (x^2 -1)^2 -1 = x^4 - 2x^2* *(g o g)(x) = (x+1) + 1 = x +2* **b. f(x) = 2x -3 and g(x) = 1/(x + 3)** *(f o g)(x) = 2(1/(x+3)) - 3 = 2/(x+3) - 3 = (2-3x-9)/(x+3) = (-3x-7)/(x+3)* *(g o f)(x) = 1/((2x-3)+3) = 1/(2x) = 1/(2x)* *(f o f)(x) = 2(2x-3) -3 = 4x - 6 -3 = 4x - 9* *(g o g)(x) = 1/((1/(x+3))+3) = 1/((1+3x+9)/(x+3)) = (x+3)/(3x+10)* 3. If h(x) = 1/(x-1)^3, and if h(x) = f(g(x)), which of these statements could be correct? * **A. f(x) = x^2, and g(x) = 1/(x-1)** * B. f(x) = x^2, and g(x) = 1/(x-3) * C. f(x) = 1, and g(x) = x-3 * **D. f(x) = 1, and g(x) = x-1** * **A and D.** are correct 4. If f(x) = √2x-1 and g(x) = x^2, the value of g(f(13)) * g(g(4)) equals 6400 ## Log Functions: 1. Given that log3 a = 6 and log3 b = 5, the value of log3 (9ab^2) is 18. 2. Evaluate the following expressions: **a. log4 16 = 2** **b. log1 1 = 0** **c. log8 8 = 1** **d. 5log2 25 (use a calculator)= 12.5** 3. Evaluate. **a. 2log1 - 5 log2 (1/8) = +15** **b. log3 (log2 (log4 64)) = 0 ** **c. log2 (log3 (1/9)) = -3** **d. √(1/4)log3√3 - 7log3 0.5 = (5/8)** **e. √2log7√49 + log7(1/√6) = -4**