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C ollege A lgebra. Functions and Graphs (Chapter1). Objectives. Cover the topics in Section ( 1-5):Graphs and Transformations. Library of Elementary Functions. Vertical and horizontal Shifts. Reflection, Expansions, and Contractions. Chapter1. Absolute Value Function.
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College Algebra Functions and Graphs (Chapter1)
Objectives Cover the topics in Section ( 1-5):Graphs and Transformations • Library of Elementary Functions. • Vertical and horizontal Shifts. • Reflection, Expansions, and Contractions. Chapter1
Absolute Value Function Identity Function g(x) f(x) 5 5 x –5 5 x –5 5 g(x) = |x| f(x) = x –5 1. 2. Graphs and Transformations Six Basic Functions 3 Chapter1
Square Function Cube Function h(x) m(x) 5 5 x –5 5 x –5 5 2 3 h(x) = x m(x) = x –5 3. 4. Graphs and Transformations Six Basic Functions 4 Chapter1
Cube-Root Function Square-Root Function p(x) n(x) 5 5 x –5 5 x 5 3 n(x) = x p(x) = x –5 5. 6. Graphs and Transformations Six Basic Functions 5 Chapter1
Graphs and Transformations Vertical Translation: Y-k = f(x) Horizontal Translation: y = f(x-h) Reflection: y = – f(x) Reflect the graph of y = f(x) in the x axis Vertical Expansion and Contraction: y = A f(x) k > 0 Shift graph of y = f(x) up k units k < 0 Shift graph of y = f(x) down k units h > 0 Shift graph of y = f(x) right h units h < 0 Shift graph of y = f(x) left h units A > 1 Vertically expand graph of y = f(x) by multiplying each ordinate value by A 0 < A < 1 Vertically contract graph of y = f(x) by multiplying each ordinate value by A 6 Chapter1
Graphs and Transformations 7 Chapter1
Graphs and Transformations Example 1: Sketch the graph of the function f(x)=|x|+1. Do not plot points, but instead apply transformations to the graph of a standard function. Solution 8 Chapter1
Graphs and Transformations Example 2: Sketch the graph of the function f(x)=|x-1|. Do not plot points, but instead apply transformations to the graph of a standard function. Solution 9 Chapter1
y g(x) = |x| + 3 8 f(x) = |x| 4 h(x) = |x| –4 x 4 -4 -4 Graphs and Transformations Example 3:Use the graph of f (x) = |x| to graph thefunctions g(x) = |x| + 3 and h(x) = |x| – 4. Example 3:Use the graph of f (x) = |x| to graph thefunctions g(x) = |x| + 3 and h(x) = |x| – 4. Solution 10 Chapter1
Graphs and Transformations Example 4: Solution 11 Chapter1
Graphs and Transformations Example 5: Solution 12 Chapter1
Graphs and Transformations Example 6: 13 Chapter1
Graphs and Transformations 14 Chapter1
y y 4 4 x x -4 -4 (–1, –2) Graphs and Transformations Example(7) Graph the function using the graph of Then a horizontal shift 5 units left. First make a vertical shift 4 units downward. (4, 2) (0, 0) (4, –2) (–5, –4) (0, – 4) 15 Chapter1
Graphs and Transformations 16 Chapter1
Graphs and Transformations 17 Chapter1
Graphs and Transformations Example(8) Sketch the graph of the function -|x| . Do not plot points, but instead apply transformations to the graph of a standard function. 18 Chapter1
Graphs and Transformations Example(9) Sketch the graph of the function 2|x| . Do not plot points, but instead apply transformations to the graph of a standard function. 19 Chapter1
Graphs and Transformations Example(10) 20 Chapter1
Graphs and Transformations 21 Chapter1
y y 4 4 x x 4 4 – 4 -4 Graphs and Transformations Example(11) Graph y = –(x + 3)2 using the graph of y = x2. Then shiftthe graphthree units to the left. First reflectthe graphin the x-axis. y = x2 (–3, 0) y =–(x + 3)2 y =–x2 22 Chapter1
y 4 is the graphof y = x2shrunk vertically by. x –4 4 Graphs and Transformations Example(12) Vertical Stretching and Shrinking If c > 1 then the graph of y= cf(x) is the graph of y = f(x) stretched vertically by c. If 0 < c < 1 then the graph of y = cf(x) is the graph of y = f(x) shrunk vertically by c. y = x2 y =2x2 Example: y =2x2 is the graph of y = x2stretched verticallyby 2. Chapter1
y 4 is the graph of y = |x| stretched horizontallyby . x -4 4 Graphs and Transformations Example(13) Horizontal Stretching and Shrinking If c > 1, the graph of y = f(cx) is the graph of y = f(x) shrunk horizontally by c. If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) stretchedhorizontally by c. y = |2x| Example:y = |2x|is the graph of y = |x| shrunk horizontally by 2. y = |x| Chapter1
5 –5 5 –5 Graphs and Transformations Example(14) Sketch the graphs given by Chapter1
y y 8 8 4 4 x x -4 4 4 Step 3: -4 Step 4: Graphs and Transformations Example(15) Graph using the graph of y = x3. Step 1: y = x3 Step 2: y =(x + 1)3 Chapter1
Graphs and Transformations Example(16) Chapter1
Graphs and Transformations Chapter1
Graphs and Transformations Chapter1
Graphs and Transformations Example(17) Chapter1
Graphs and Transformations Chapter1
Graphs and Transformations Chapter1
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