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Chapter 5

Chapter 5. Quadratic Equations and Functions. 5-1 Warm Up. What is a quadratic equation? What does the graph look like? Give a real world example of where it is applied. 5-1 Modeling Data with Quadratic Functions. OBJ: Recognize and use quadratic functions

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Chapter 5

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  1. Chapter 5 Quadratic Equations and Functions

  2. 5-1 Warm Up • What is a quadratic equation? What does the graph look like? Give a real world example of where it is applied.

  3. 5-1 Modeling Data with Quadratic Functions OBJ: Recognize and use quadratic functions Decide whether to use a linear or a quadratic model

  4. Quadratic Functions • Quadratic function is a function that can be written in the form f(x)= ax²+bx+c, where a ≠ 0 • The graph is a parabola • The ax² is the quadratic term • The bx is the linear term • The c is the constant term

  5. The highest power in a quadratic function is two • A function is linear if the greatest power is one

  6. Tell whether each function is linear or quadratic • F(x)= (-x+3)(x-2) • Y=(2x+3)(x-4) • F(x)=(x²+5x)-x² • Y= x(x+3)

  7. Modeling Data • Last semester you modeled data that, when looking at the scatter plot, the data seemed to be linear • Some data can be modeled better with a quadratic function

  8. Find a quadratic model that fits the weekly sales for the Flubbo Toy Comp

  9. 5-1 Wrap Up • What is a quadratic function? • What kinds of situations can a quadratic function model?

  10. 5-2 Warm Up List as many things as you can that have the shape of a parabola

  11. 5-2 Properties of Parabolas OBJ: Find the min and max value of a quadratic function Graph a parabola in vertex form

  12. Comparing Parabolas • Any object that is tossed or thrown will follow a parabolic path. • The highest or lowest point in a parabola is the vertex • It is the vertex that is the maximum or minimum value • If a is positive the parabola opens up, making the vertex a min point • If a is negative the parabola opens down, making the vertex a max point

  13. Axis of symmetry divides a parabola into two parts that are mirror images of each other • The equation of the axis of symmetry is x=(what ever the x coordinate of the vertex is) • Two corresponding points are the same distance from the axis of symmetry

  14. Y=ax²+bx+c is the general equation of a parabola • If a>0 the parabola opens up • If a<0 the parabola opens down • If a is a fraction it is a wide opening • If a is a whole number it is narrow

  15. Examples • Each tower of the Verrazano Narrows Bridge rises about 650 ft above the center of the roadbed. The length of the main span is 4260 ft. Find the equation of the parabola that could model its main cables. Assume that the vertex of the parabola is at the origin.

  16. Translating Parabolas • Not every parabola has its vertex at the origin • Y=a(x-h)²+k is the vertex form of a parabola • It is a translation of y=ax² • (h,k) are the coordinates of the vertex

  17. Sketch the following graphs

  18. The vertex is ( h, k) Axis of symmetry is x = h If a > 0 it is a max If a < 0 it is a min

  19. Graph, give equation for axis of symmetry and state the vertex.

  20. Example • Sketch the graph of y= -1/2(x-2)²+3 • Sketch the graph of y = 3(x+1)²-4

  21. Wrap Up 5-2 • What does the vertex form of a quadratic function tell you about its graph?

  22. Warm Up 5-3 • List formulas that you know to use to find answers to problems quickly. (list as many formulas as you can)

  23. 5-3 Comparing Vertex and Standard Forms OBJ: Find the vertex of a function written in standard form Write equations in vertex and standard form

  24. Get into a group of four • Turn to page 211 • Do part a • What do you notice about the graphs of each pair of equations? • What is true of each pair of equations? • Write a formula for the relationship between b and h • How can we modify our formula to show the relationship among a, b, and h. (the last couple of equations)

  25. Standard form of a parabola • When a parabola is written as y=ax²+bx+c it is standard form • The x coordinate of the vertex can be found by –b/(2a) • To find the y coordinate by – [(b^2-4ac)/4a]

  26. Suppose a toy rocket is launched to its height in meters after t seconds is given by H = -4.9t^2 +20t +1.5. How high is the rocket after one second? How high is the rocket when launched. How high is the rocket after 12 seconds?

  27. Example • Write the function y= 2x²+10x+7 in vertex form • Write the function y= -x²+3x-4 in vertex form • What is the relationship between the axis of symmetry and the vertex of the parabola?

  28. Example • As a graduation gift for a friend, you plan to frame a collage of pictures. You have a 9 ft strip of wood for the frame. What dimensions of the frame give you maximum area of the collage? • What is the maximum area for the collage? • What is the best name for the geometric shape that gives the maximum area for the frame? • Will this shape always give the max area?

  29. Consider this general formula:

  30. A ball is dropped form the top of a 20 meter tall building. Find an equation describing the relation between the height and time. Graph its height h after t seconds. Estimate how much time it takes the ball to fall to the ground. Explain your reasoning

  31. Write y= 3(x-1)²+12 in standard form • A rancher is constructing a cattle pen by a river. She has a total of 150 ft of fence, and plan to build the pen in the shape of a rectangle. Since the river is very deep, she need only fence three sides of the pen. Find the dimensions of the pens so that it encloses the max area.

  32. Suppose a swimming pool 50 m by 20 m is to be built with a walkway around it. IF the walkway is w meters wide, write the total area of the pool and walkway in standard form

  33. Consider this If a quarterback tosses a football to a receiver 40 yards downfield, then the ball reaches a maximum height halfway between the passer and the receiver, it will have a equation

  34. Example Suppose a defender is 3 yards in front of the receiver. This means the defender is 37 yards from the quarterback. Will he be able deflect or catch the ball?

  35. Examples A model rocket is shot at an angle into the air from the launch pad. The height of the rocket when it has traveled horizontally x feet from the launch pad is given by

  36. A 75-foot tree, 10 feet from the launch pad is in the path of the rocket. Will the rocket clear the top of the tree? Estimate the maximum height the that the rocket will reach.

  37. Wrap Up 5-3 • Describe the similarities and differences between the vertex form and standard form of quadric equations.

  38. Warm Up 5-4 • Name mathematical operations that are opposites of each other. For example, addition is the opposite of subtraction. • Two inverse functions are opposite of each other in the same way.

  39. 5-4 Inverses and Square Root Functions OBJ: Find the inverse of a function Use square root functions

  40. Consider the functions • F(x)= 2x-8 • G(x)= (x+8)/2 • Find F(6) and G(4) • F(x) and G(x) are inverses because one function undoes the other • Graph each function on the same coordinate plane • Find three coordinates on f(x) • Reverse the coordinates and graph • What do you notice?

  41. Definition The inverse of a relations is the relation obtained by reversing the order of the coordinates of each ordered pair in the relation

  42. Remember • If the graph of a function contains a point (a,b), then the inverse of a function contains the point (b,a)

  43. Example

  44. Inverse Relation Theorem Suppose f is a relation and g is the inverse of f. Then: • A rule for g can be found by switching x and y • The graph of g is the reflection image of the graph of f over y=x • The domain of g is the range of f, and the range of g is the domain of f

  45. Remember • The inverse of a relation is always a relation • The inverse of a function is not always a function

  46. Examples Consider the function with equation y= 4x-1. Find an equation for its inverse. Graph the function and its inverse on the same coordinate plane. Is the inverse a function?

  47. Consider the function with domain the set of all real numbers and equation y=x^2 What is the equation for the inverse? Graph the function and its inverse on the same coordinate plane. Is the inverse a function? Why or Why not?

  48. Example • Graph the function and its inverse. The write the equation of the inverse

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