Chapter 3.4 Graphs and Transformations

By Saho Takahashi

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.

Parent Functions

Parent Functions

Parent Functions

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.

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.

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.

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.

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.

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.

Compression and Stretches

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

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.

Example 4Q. Graph g(x) =2³and h(x) =1/4x³Answer: 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.

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.

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.

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 function3. Graph g(x) by shifting 4 left from the parent function.

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) = x3

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

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).

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) = x2 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.

Word Problem 5Q. 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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