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University of Palestine IT-College. C ollege A lgebra. Functions and Graphs (Chapter1) L:9. Objectives. Cover the topics in Section ( 1-5):Graphs and Transformations. Library of Elementary Functions. Vertical and horizontal Shifts. Reflection, Expansions, and Contractions. Chapter1.
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University of Palestine IT-College College Algebra Functions and Graphs (Chapter1) L:9
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
End of the Lecture Let Learning Continue