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Chapter 5

Chapter 5. Quadratic Equations and Functions. In This Chapter You Will …. Learn to use quadratic functions to model real-world data. Learn to graph and to solve quadratic equations. Learn to graph complex numbers and to use them in solving quadratic equations.

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Chapter 5

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  1. Chapter 5 Quadratic Equations and Functions

  2. In This Chapter You Will … • Learn to use quadratic functions to model real-world data. • Learn to graph and to solve quadratic equations. • Learn to graph complex numbers and to use them in solving quadratic equations.

  3. 5.1 Modeling Data With Quadratic Functions • 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem. What you’ll learn … • To identify quadratic functions and graphs • To model data with quadratic functions

  4. Quadratic Functions and their Graphs • A quadratic function is a function that can be written in the standard form, where a≠0. f(x) = ax2 + bx + c Quadratic term Linear term Constant term

  5. y = (2x +3)(x – 4) f(x) = 3(x2-2x) – 3(x2 – 2) Example 1 Classifying Functions Determine whether each function is linear or quadratic. Identify the quadratic, linear, and constant terms.

  6. The graph of a quadratic function is a parabola. The axis of symmetry is the line that divides a parabola into two parts that are mirror images.

  7. The vertex of a parabola is the point at which the parabola intersects the axis of symmetry. The y-value of the vertex represents the maximum or minimum value of the function.

  8. Vertex ___________ Axis of Symmetry ______ Example 2a Graph y = 2x2 – 8x + 8

  9. Vertex ___________ Axis of Symmetry ______ Example 2b Graph y = -x2 – 4x + 2

  10. Find a quadratic function to model the values in the table. Substitute the values of x and y into y = ax2 + bx + c. The result is a system of three linear equations. Example 3a Finding a Quadratic Model

  11. Find a quadratic function to model the values in the table. Substitute the values of x and y into y = ax2 + bx + c. The result is a system of three linear equations. Example 3b Finding a Quadratic Model

  12. Example 4 Real World Connection The table shows the height of a column of water as it drains from its container. Model the data with a quadratic function. Graph the data and the function. Use the model to estimate the water level at 35 seconds. Step 1 Enter data into L1 and L2. Use QuadReg. Step 2Graph the data and the function. Step 3 Use the table to find f(35).

  13. 5.2 Properties of Parabolas 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem. What you’ll learn … • To graph quadratic functions • To find maximum and minimum values of quadratic functions

  14. Graphing Parabolas • The standard form of a quadratic function is y=ax2 + bx + c. When b=0, the function simplifies to y=ax2 + c. • The graph of y=ax2 + c is a parabola with an axis of symmetry x =0, the y-axis. The vertex of the graph is the y-intercept (0,c).

  15. Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. • When a>0, the parabola opens up. • When a<0, the parabola opens down.

  16. Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. • The axis of symmetry is x= - b 2a

  17. Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. • The vertex is ( - , f(- ) ). b 2a b 2a

  18. Properties Graph of a Quadratic Function in Standard Form The graph of y=ax2 + bx + c is a parabola when a≠0. • The y intercept is (0,c).

  19. Quadratic Graphs 2 y = x The graph of a quadratic function is a U-shaped curve called a parabola. .

  20. Graph y= -½x2 + 2 Graph y= 2x2 - 4 Example 1 Graphing a Function of the Form y=ax2 + c

  21. Symmetry 2 y = x + 3 You can fold a parabola so that the two sides match evenly. This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry.

  22. Vertex The highest or lowest point of a parabola is its vertex, which is on the axis of symmetry. 2 y = ½ x y = -4 x +3 2 Minimum Maximum

  23. Determining Vertex and Axis of Symmetry 2 2 2 2

  24. 5.3 Translating Parabolas What you’ll learn … • To use the vertex form of a quadratic function 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem.

  25. Investigation: Vertex Form b 2a

  26. In other words … To translate the graph of a quadratic function, you can use the vertex form of a quadratic function.

  27. Properties The graph of y = a(x – h)2 + k is the graph of y = ax2 translated h units horizontally and k units vertically. • When h is positive the graph shifts right; when h is negative the graph shifts left. • When k is positive the graph shifts up; when the k is negative the graph shifts down. • The vertex is (h,k) and the axis of symmetry is the line x=h.

  28. Graph y = - (x-2)2 +3 Graph the vertex. Draw the axis of symmetry. Find another point. When x=0. Sketch the curve. Example 1a Using Vertex Form to Graph a Parabola 1 2

  29. Graph y = 2 (x+1)2 - 4 Graph the vertex. Draw the axis of symmetry. Find another point. When x=0. Sketch the curve. Example 1b Using Vertex Form to Graph a Parabola

  30. Example 2a Writing the Equation of a Parabola • Write the equation of the parabola. • Use the vertex form. • Substitute h=__ and k= ___. • Substitute x=0 and y = 6. • Solve for a.

  31. Example 2b Writing the Equation of a Parabola • Write the equation of the parabola. • Use the vertex form. • Substitute h=__ and k= ___. • Substitute x=___ and y = ___. • Solve for a.

  32. Example 2c Writing the Equation of a Parabola • Write the equation of a parabola that has vertex (-2, 1) and goes thru the point (1,28). • Write the equation of a parabola that has vertex (-1, -4) and has a y intercept of 3.

  33. y = 2x2 +10x +7 y = -3x2 +12x +5 Convert to Vertex Form

  34. y = (x+3)2 - 1 y = -3(x -2 )2 +4 Convert to Standard Form

  35. Example 3 Real World Connection • The photo shows the Verrazano-Narrows Bridge in New York, which has the longest span of any suspension bridge in the US. A suspension cable of the bridge forma a curve that resembles a parabola. The curve can be modeled with the function y = 0.0001432(x-2130)2 where x and y are measured in feet. The origin of the function’s graph is at the base of one of the two towers that support the cable. How far apart are the towers? How high are they?

  36. Start by drawing a diagram. • The function, y = 0.0001432(x-2130)2 , is in vertex form. Since h =2130 and k =0, the vertex is (2130,0). The vertex is halfway between the towers, so the distance between the towers is 2(2130) ft = 4260 ft. • To find the tower’s height, find y for x=0.

  37. 5.4 Factoring Quadratic Expressions • What you’ll learn … • To find common and binomial factors of quadratic expressions • To factor special quadratic expressions 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

  38. Investigation: Factoring * • Since 6 3 = 18, 6 and 3 up a factor pair for 18. • Find the other factor pairs for 18, including negative integers. • Find the sum of the integers in each factor pair for 18. • Does 12 have a factor pair with a sum of -8? A sum of -9? • Using all the factor pairs of 12, how many sums are possible? • How many sums are possible for the factor pairs of -12?

  39. Factoring is rewriting an expression as the product of its factors. • The greatest common factor (GCF) of an expression is the common factor with the greatest coefficient and the greatest exponent.

  40. 4x + 12 x - 8 GCF ________ 4b -2b -6b GCF ________ Example 1a Finding Common Factors 3 2 3 2

  41. 3x - 12x +15x ( ) 6m - 12m - 24m ( ) Example 1b Finding Common Factors 3 2 3 2 GCF GCF

  42. Factor x2 +8x +7 Factor x2 +6x +8 Example 2 Factoring when ac>0 and b>0 Factor x2 +12x +32 Factor x2 +14x +40

  43. Factor x2 -17x +72 Factor x2 -6x +8 Example 3 Factoring when ac>0 and b<0 Factor x2 -7x +12 Factor x2 -11x +24

  44. Factor x2 - x - 12 Factor x2 +3x - 10 Example 4 Factoring when ac<0 Factor x2 -14x - 32 Factor x2 +4x - 5

  45. Factor 2x2 +11x + 12 Factor 3x2 - 16x +5 Example 5 Factoring when a≠0 and ac>0 Factor 4x2 +7x + 3 Factor 2x2 - 7x + 6

  46. Factor 4x2 -4x - 15 Factor 2x2 +7x - 9 Example 6 Factoring when a≠0 and ac<0 Factor 3x2 - 16x - 12 Factor 4x2 +5x - 6

  47. Special Cases • A perfect square trinomial is the product you obtain when you square a binomial. • An expression of the form a2 - b2 is defined as the difference of two squares.

  48. x - 8x + 16 n - 16n + 64 Factoring a Perfect Square Trinomial with a = 1 2 2

  49. x - 121 ( ) ( ) The Difference of Two Squares 4x - 36 ( ) ( ) 2 2

  50. 5.5 Quadratic Equations What you’ll learn … • To solve quadratic equations by factoring and by finding square roots • To solve quadratic equations by graphing 2.02 Use quadratic functions and inequalities to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem.

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