UNIT 1: QUADRATIC FUNCTIONS UNIT 1 By the end of this unit, I can Draw the graph of a function using different methods Explain the meaning of the term function and distinguish between a function and a relation and substitute and evaluate linear and quadratic functions represented by function notation Explain the meanings of the terms domain and range and explain restrictions on the domain and range of a function Sketch a graph of f(x)=x 2, and describe its key properties Describe and graph the translations, stretches and reflections of quadratic functions Solve real-world problems involving quadratic functions Name: 1
1.0 Introduction to Quadratic Functions Success Criteria: I can o Draw the graph of a function using different methods Quadratic Functions 2
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1.1 - WHAT IS A FUNCTION? Success Criteria: I can Explain the meaning of the term function and distinguish between a function and a relation Relation: Function: EX 1: Given a graph of a relation, how can we tell if the relation is function? Relation A: Relation B: Relation C: Relation D: EX 2: Given a table of values of a relations, how can we tell if the relation is function? Relation E: Relation F: x y x Y -3 4 0 6-2 6 1 5-1 9 2 4 0 2 3 3 1 1 2 2 5
EX 3: Given a mapping diagram of a relation (mapping diagram: a visual way of representing a relation, where arrows are used to map the x-values onto their corresponding y-values), how can we tell if the relation is function? Relation G: Relation H: -2-1 0 1 2 7 6 5 4 3-5 0-5 0 5 EX 4 Given an equation, how can we tell if the relation is a function? Relation I Relation J Relation K y = 4x + 6 y = x ' + 8x 6 x ' + y ' = 25 In Summary: 6
Practice Determine if each relation is a function or relation: Relation A {(-5, 4), (0, -3), (2, -1), (2, -5), (4, 10)} Relation B x y -2 5-1 5 0-2 1 4 2 0 Function or Relation Why? Function or Relation Why? Relation C Relation D 1 2 3 4-4 0 5 Function or Relation Why? Function or Relation Why? Relation E Relation D Function or Relation Why? Function or Relation Why? Homework: 7
1.2 FUNCTION NOTATION Success Criteria: I can Explain the meaning of the term function and distinguish between a function and a relation Note! An equation that is a function can be represented using function notation. In function notation, the symbol f (x) is another name for y and represents the value of the function f at x. We write the function y = 3x + 2 in function notation as f(x) = 3x + 2 ***y and f(x) are the same!*** EX: 1: If f (x) = 3 x + 1, find a) f (6) b) f ( 2) c) f (0) EX 2: If h(x) = x 2 + x - 9, find h(8). EX 3: If g(x) = -3x + 10, find the value of x when g(x) = 28. EX 4: If f(x) = x 2 +20, find the value of x when f(x) = 24. 8
EX 5: Given the function f(x)= -(x-2)2-3, determine the following: a) Evaluate f(0). b) What does f(0) represent on the graph? c) Does f(2)=3? Explain. d) If f(x)=-4, determine the possible value(s) of x. Homework: 9
1.1 1.2 Practice 1. Determine whether the following are relation or functions. If they are not functions, state why. 3. State whether each of the following equations are functions or relations. a. y = 5x 2x ' b. 8 = x + 6y c. x = 7y ' + 1 d. 5 x + 1 ' = 86 2. Use the vertical line test to determine whether the following are relation or functions. e. x ' + y ' = 44 4. Determine whether the following are relations or functions. a) The relation between earnings and sales if Lala earns $400 per week plus 5% commission on sales. b) The relation between distance and time if Ken walks at 5 km/h c) The relation between students ages and the number of credits earned. 10
5. Evaluate the following, where f x = 2 3x a) f( 2) b) f( 2 ' ) 6. For g x = 4 5x, determine the value of x when g x is equal to: a) -6 b) 2 7. The graphs of f x and g(x) are shown below. 11
1.3 DOMAIN AND RANGE Success Criteria: I can Explain the meanings of the terms domain and range Restrict the domain and range of a function Domain: Range: Recall from lesson 1.1: A function can be defined as a set of ordered pairs in which for each element in the domain, there is exactly one element in the range. EX 1: Determine the domain and range. Method 1: List the numbers Method 2: Describe in Words 9 8 7 6 5 4 3 2 1 y x Method 3: Write as set notation 4 3 2 1 1 1 2 3 4 5 2 3 RECALL: Less than < 2 <3 2 is less than 3 Greater than > 5>4 5 is greater than 4 Less and equal x 4 x is less and equal to 4 Greater and equal x 6 x is greater than and equal to 6 12
EX 2: Determine the domain and range in set notation. 9 8 7 6 5 4 3 2 1 y x 4 3 2 1 1 1 2 3 4 5 2 3 Practice: Determine the domain and range for the following functions: a) Domain: 5 y 4 3 2 1 x 4 3 2 1 1 1 2 3 4 5 2 b) Domain: 3 5 y 4 3 2 1 x 4 3 2 1 1 1 2 3 4 5 2 c) Domain: 3 Range: Range: Range: EX 3: A function represents the relationship between the height of a ball and the time elapsed since the ball was thrown. The ball reaches a maximum height of 20 m after 1.5 s. The ball hits the ground at 3 s. State a reasonable domain and range for the relation 13
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Domain and Range Extra Practice For the following relations, identify the domain and range. 1. 2. 3.. 4. 5. 6. 15
7. y = 3x ' 8. {(5, 2), ( 3, 1), (2, 4), (0, 11)} 9. 10. x y 0 12 1 13 2 15 3 14 11. A scenario where time versus height is measured. You kick a ball and it lands 4 seconds later. The highest it goes is 8 m. 12. f x = 9x + 1 16
1.4 - QUADRATIC FUNCTIONS Success Criteria: I can Sketch a graph of f(x)=x 2, and describe its key properties Equation Examples Linear Quadratic Graph Examples Another way to classify if a function is linear or quadratic is to calculate the 1 st and 2 nd differences: Linear functions will have constant differences Quadratic functions will have constant differences EX1: Classify each as a linear or quadratic function. x y First Differences Second Differences x y First Differences Second Differences 0 3-2 -3 1 6-1 -4 2 9 0-3 3 12 1 0 4 15 2 5 17
The simplest quadratic function is the parabola, y = x 2. EX2: Recall the following key properties of a parabola: Vertex Axis of Symmetry Maximum/ Minimum Value x-intercepts y-intercept EX 3: A community centre is building a new fence to surround a play area. The area to be enclosed, A(x) in square metres, is modeled by a quadratic function, A(x) = x 15 ' + 225, where x is the width of the play area in metres. a) How can we tell from the equation that the function is quadratic? b) Complete the table of values for x = 0, 5, 10, 15, 20, 25, 30 x 0 A(x) 5 10 15 20 25 30 c) Graph the function using the table of values. d) Identify the vertex, max/min values and intercepts of the parabola. Homework: 18
1.5 STRETCHES AND REFLECTIONS of y=x 2 Success Criteria: I can Describe and graph the translations, stretches and reflections of quadratic functions Today, we will be looking at quadratic equations of the following form: y = ±ax ' This represents two types of transformation on our parent function y = x ' : reflection on the x-axis vertical stretch/compression EX 1: Graph the following functions on the same grid below. Use table of values: a) y = x ' b) y = 2 ' x' c) y = 2x ' x y x y x y -3-3 -3-2 -2-2 -1-1 -1 0 0 0 1 1 1 2 2 2 3 3 3 Describe how each graph compares to the graph of a) y=x 2. 19
SUMMARY For a quadratic function of the form f(x)= ax 2, the value of a determines if the graph represents vertical stretch, compression or a reflection in the x-axis to the graph of f(x)=x 2. If a > 1, the graph is a: If 0>a>1, the graph is a: If a<0, the graph is: EX 2: Practice a) Describe the transformations that have taken place for each function. b) Graph each parabola with the parent function y=x 2. i) y = 3x ' ii) y = 2 ' x' 20
EX 3: Write an equation of a parabola that satisfies each set of conditions: EX 4: The graph of f(x)=ax 2 is shown. What is the value of a? Homework: 21
1.6 TRANSLATIONS of y=x 2 Success Criteria: I can Describe and graph the translations, stretches and reflections of quadratic functions Homework 22
1.7 COMBINATIONS of transformations Success Criteria: I can Describe and graph the translations, stretches and reflections of quadratic functions Homework 23