Graphing Functions Using Transformations

Let’s address the questions based on the provided image content. Question 1 a) g is related to one of the six parent functions. (c) Identify the parent function f. (b) Describe the sequence of transformations from f to g. (c) Sketch the graph of g by hand. (d) Use function notation to write g in terms of the parent function f, g(x) = ⅓ f(x - 2) + 1. Answer: a) Parent role f(x) = x², b) Vertical shrinkage by a factor of ⅓, Horizontal shift to the right by 2 units of ⅓, c) The graph will be: [Graph of g(x) showing a vertically shrunk and right-shifted parabola] d) When written in function notation, g(x) = ⅓ f(x - 2) Question 2 Sketch a graph of the function. Use transformations of functions when ever possible. f(x) = ⅓ (x - 5), 2 ≤ x ≤ 8 Answer: Step 1: Shown is the graph of f(x) = ⅓ (x - 5) Step 2: 2 ≤ x ≤ 8 is on the horizontal axis and y is on the vertical axis. [Graph of f(x) = ⅓ (x - 5) from x = 2 to x = 8] Question 3 Begin by graphing f(x) = log₂x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each domain and range. g(x) = -2 log₂(x) Answer: Step 1: For the following logarithmic function defined by 1) f(x) = log₂x. Graphing logarithmic function given in equation (1) requires setting up table of coordinates, so that | x | f(1) = y = log₂(1) = 2⁰ = 1, 2⁰ = 1 → y = 0 | (1, 0) | | 2 | f(2) = y = log₂(2) = 2¹ = 2, 2¹ = 2 → y = 1 | (2, 1) | | 4 | f(4) = y = log₂(4) = 2² = 4, 2² = 4 → y = 2 | (4, 2) | | 8 | f(8) = y = log₂(8) = 2³ = 8, 2³ = 8 → y = 3 | (8, 3) | | 16 | f(16) = y = log₂(16) = 2⁴ = 16, 2⁴ = 16 → y = 4 | (16, 4) | | 32 | f(32) = y = log₂(32) = 2⁵ = 32, 2⁵ = 32 → y = 5 | (32, 5) | | 64 | f(64) = y = log₂(64) = 2⁶ = 64, 2⁶ = 64 → y = 6 | (64, 6) | Step 2: We plot the following points between (x, y) determined from the table of coordinates and connect them with the continuous curve which represent the graph of logarithmic function f(x) = log₂x as shown in figure (1). [Figure 1 shows the graph of f(x) = log₂x] Step 3: Note that: The y-axis which represented by the equation x = 0 is the vertical asymptote, so that the curve approaches, but never touches the positive portion of the y-axis as shown in figure (1). The domain of f(x) = log₂(x) is all positive real numbers x ∈ (0, ∞) and the range is all real numbers y ∈ (-∞, ∞). Step 4: To graph g(x), the logarithmic function g(x) = -2 log₂(x) transform to a graph g(x) = -2 log₂(x) then the graph g(x) = -2 log₂(x) transform to a graph g(x) = -2 log₂(x) so that table o coordinates becomes Question 4 Use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function’s equation and transformations of f(x) = ½x to graph g(x) = (2x + 7)/(2x + 6) Answer: Step 1: g(x) = (2x + 7)/(2x + 6) | 2x + 6 | 2x + 7 |___| 2x | 2x + 6 (2x + 6) | - (2x + 6) ___| Remainder is 1 The quotient is 1 and the remainder is 1/(2x + 6) Step 2: g(x) = 1 + 1/(2x + 6). We can graph g by horizontal shifting 3 units left then vertical shifting 2 units up. [Graph of g(x) = 1 + 1/(2x + 6)] Question 5 What transformations of the parent graph of f(x) = √x to graph g(x) = -√(12x + 4) Answer: Step 1: First we are going to graph the parent function and the first transformation and second transformation of parent function. [Graph showing parent function √x and transformations] Step 2: We can see from the graph the first transformation of parent function has horizontal shift of 10 units down. Vertical stretch was also performed on the first function, and vertical shift of 4 units up. Horizontal compression was also performed on the first function. [Final graph of g(x) = -√(12x + 4)]