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Chapter 3.4 Graphs and Transformations By Saho Takahashi

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Parent Functions Parent Function – is a function with a certain shape that has the simplest algebraic rule for that shape. The parent functions are used to illustrate the rules for the basic transformations.

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Parent Functions

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Vertical Shifts Let c as a positive number. The graph of g(x) = f(x) + c is the graph of f shifted upward c units. The graph of g(x) = f(x) – c is the graph of f shifted downward c units.

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Example 1 - Shifting a graph Vertically Q. Graph g(x) = |x|+ 4, and h(x) = |x|- 3 Answer : The parent function of g(x) and h(x) is f(x) = |x|. The graph of g(x) is the graph of f(x) = |x| shifted 4 units upward, and the graph of h(x) is the graph of f(x) = |x| shifted 3 units downward.

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Horizontal Shifts Let c be a positive number. The graph of g(x)= f(x + c) is the graph of f shifted c units to the left. The graph of g(x) = f(x - c) is the graph of f shifted c units right.

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Example 2 – Shifting a Graph Horizontally Q. Graph g(x) = 1/ x – 3 and h(x) = 1/ x + 4. Answer : the parent function of g(x) and h(x) is f(x) = 1/ x. The graph of g(x) is the graph of f(x) = 1/ x shifted 3 units to the right, and the graph of h(x) = 1 / x shifted 4 units to the left.

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Reflections The graph of g(x) = - f(x) is the graph of f reflected across the x – axis. The graph of g(x) = f( -x ) is the graph of f reflected across the y – axis.

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Example 3 – Reflecting a graph Across the x – or y – axis. Graph g(x) = - [x] and h(x) = [-x] Answer: The parent function of functions g(x) and h(x) is f(x) = [x]. the graph of g(x) is the graph of f(x) = [x] reflected across the x – axis, and the graph of h(x) is the graph of f(x) = [x] reflected across the y – axis.

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Compression and Stretches Graphs of function may be either compressed or stretched vertically and horizontally.

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Vertically Stretches and Compressions In the function of the graph g(x) =c· f(x) c > 0…. If c > 1, the graph of g(x) = c· f(x) is the graph of f stretched vertically, away from the x – axis, by a factor of c. If c < 1, the graph of g(x) = c· f(x) is the graph of f compressed vertically, toward the x – axis, by a factor of c.

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Example 4 Q. Graph g(x) =2³and h(x) =1/4x³ Answer: The parent function is f(x) = x³. For the graph of g(x), every y coordinate of the parent function is multiplied by 2, stretching the graph of the function in the vertical direction away from the x axis. For the function h(x)m every y – coordinate of the parent function is multiplied by ¼, compressing the graph of the function in vertical direction toward the x – axis.

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Horizontal Stretches and Compressions In the function of the graph of g(x) = f( c· x ). If c > 1, the graph of g(x) = f( c· x ) is the graph of f compressed horizontally toward the y axis by a factor of 1/c. If c < 1, the graph of g(x) = f( c· x ) is the graph of f stretched horizontally toward the y axis by a factor of 1/c.

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Example 5 Q. Graph h(x) = [2x]. Answer: The parent function of h(x) is f(x) = [x]. The graph of the function h(x) compressed horizontally, toward the y – axis, by a factor of 1/2.

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Word Problem 1 Q. Graph function g(x) = √x+4 and its parent function on the same set of axis. Answer: 1. Find the parent function of g(x): f(x) = √x 2. Graph the parent function 3. Graph g(x) by shifting 4 left from the parent function.

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Word Problem 2 Q. Write a rule for the function whose graph can be obtained from the given parent function by performing the given transformations. Parent function: f(x) = x 3 Transformations: Shift the graph 5 units to the left and upward 4 units. A.1. Find a function that is shifted 5 units to the left: g(x) =(x+5) 3 2. Find a function that is shifted 5 units to the left and upward 4 units: g(x) = (x+5) 3 +4

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Word Problem 3 Q. Describe a sequence of transformations that transform the graph of the parent function f into the graph of the function g. f(x) = x g(x) = -3(x-4)+1 A.1. Find how much did the f(x) shift in order to be g(x): Shift f(x) 4 units to the right and 1 units upward. 2. Find how much did the f(x) reflect in order to be g(x): Reflect f(x) across the x-axis. 3. Find how did the f(x) stretched: f(x) stretched vertically by a factor of 3 in order to be g(x).

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Word Problem 4 Q. Graph each function and its parent function on the same graph. f(x) = -2(x - 1) 2 - 3 A.1. Find the parent function of f(x): f(x) = x 2 2. Find shifting: shift 1to right, and shift 3 downward. 3. Find the reflection: reflect vertically across x-axis. 4. Find the stretching and contraction: contract by a factor of 2.

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Word Problem 5 Q. In 2002, the cost of sending first-class mail was $3.07 for a letter weighing less than 1 ounce, $0.60 for a letter weighting at least one ounce, but less than 2 ounces, $0.83 for a letter weighting at least 2 ounces, but less than 3 ounces, and so on. Write a function c(x) that gives the cost of mailing a letter weighting x ounces, and graph it. A. 1.Find the function: c(x) = 23[x] +37 2.Find parent function: f(x) = [x] 3.Find shifting, stretching, contraction, and reflection. 4.Shift 37 upward, and contract by a factor of 23.

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