University:
Washington State University
Course:
MATH 172 | Calculus II
Academic year:

2024
Views:

146

Pages:

3
Author:

Bailey Ayala

Question 1 Begin by graphing $ f(x) = 2^x $. The given function can then be graphed by using transformations on this graph. Make sure to graph the asymptotes and provide their equations. To determine each function, use the graphs. Answer: Step 1: The graph of $ y = 2^x $ is shown as a dotted curve. Step 2: The graph of $ 2^{x+2} $ is the graph of $ 2^x $ translated left by two units. The asymptote has equation $ y = 0 $. [Graph of $ 2^{x+2} $ showing the horizontal asymptote at $ y = 0 $] Question 2 h is related to one of the six parent functions. a) Identify the parent function f. b) Describe the sequence of transformations from f to h. c) Sketch the graph of h by hand. d) Use function notation to write h in terms of the parent function f. $ h(x) = -(x)^2 - 8 $ Answer: a) Parent function: $ f(x) = x^2 $ b) Reflection in the y-axis, vertical shift 8 units downward. c) [Graph of $ h(x) = -(x)^2 - 8 $, a downward-opening parabola shifted 8 units down] d) In function notation: $ h(x) = -f(x) - 8 $ Question 3 Graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function’s domain and range. Answer: Both graphs are shown to the left. The graph of $ g(x) $ is a translation to the right by 1 of the graph of $ f(x) $. Both share a horizontal asymptote $ y = 0 $. The domain of both functions is $ \mathbb{R} $ and range of both functions is $ (0, \infty) $. [Graph showing $ f(x) $ and $ g(x) $, with horizontal asymptote $ y = 0 $] Question 4 Find the quadratic function that includes the set of values below: (0,5), (2,15), (4,9). The equation of the parabola is $ y = \ldots $ Answer: Let $ y = ax^2 + bx + c $ be the equation of the parabola. Set up equations: Using (0,5): $ 5 = a(0)^2 + b(0) + c $, so $ c = 5 $ Using (2,15): $ 15 = a(2)^2 + b(2) + 5 $, so $ 4a + 2b = 10 $ Using (4,9): $ 9 = a(4)^2 + b(4) + 5 $, so $ 16a + 4b = 4 $ Simplify: $ 4a + 2b = 10 $ → $ 2a + b = 5 $ (1) $ 16a + 4b = 4 $ → $ 4a + b = 1 $ (2) Subtract (1) from (2): $ (4a + b) - (2a + b) = 1 - 5 $ $ 2a = -4 $ $ a = -2 $ Substitute $ a = -2 $ into (1): $ 2(-2) + b = 5 $ $ -4 + b = 5 $ $ b = 9 $ So, the equation is $ y = -2x^2 + 9x + 5 $.