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1 Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function =

2 Eample 1: Make a table, graph and compare the graphs of = 3 and = = Compared to = Vertical: Stretch or Shrink Reflection: Yes or No Translation: Up or Down or Neither Domain: Range: Eample : Make a table, graph and compare the graphs of 1 4 and = Compared to = Vertical: Stretch or Shrink Reflection: Yes or No Translation: Up or Down or Neither Domain: Range:

3 Eample 3: Make a table, graph and compare the graphs of 1 and = Compared to = Vertical: Stretch or Shrink Reflection: Yes or No Translation: Up or Down or Neither Domain: Range: Eample 4: Make a table, graph and compare the graphs of = and = = Compared to = Vertical: Stretch or Shrink Reflection: Yes or No Translation: Up or Down or Neither 3 Domain: Range:

4 Section 10.: Graph = a + b + c PROPERTIES OF THE GRAPH OF A QUADRATIC FUNCTION To graph a parabola in the form = a + b + c : Step 1: Determine if the parabola opens up or down. If the a-value is, it opens. If the a-value is, it opens. Step : Find the ais of smmetr (the mirror of the parabola). Step 3: Find and plot the verte (middle point of the parabola). Step 4: Find more points on the graph b getting the calculator table. Plot the points. (Center the verte) Verte Other Important Information About Parabolas Minimum value Maimum value -Intercept Roots 4

5 Eample 1: Graph = Step 1: Determine whether the parabola opens up or down Step : Find and draw the ais of smmetr: Step 3: Find and plot the verte. Step 4: Find more points on the graph. Choose -values less than and greater than the verte. Verte Eample : Graph = Step 1: Determine whether the parabola opens up or down Step : Find and draw the ais of smmetr: Step 3: Find and plot the verte. Step 4: Find more points on the graph. Choose -values less than and greater than the verte. 5 Verte

6 Eample 3: Graph = Step 1: Determine whether the parabola opens up or down Step : Find and draw the ais of smmetr: Step 3: Find and plot the verte. Step 4: Find more points on the graph. Choose -values less than and greater than the verte. Verte Eample 4: The cables between two telephone poles can be modeled b the equation = , where and are measured in feet. To the nearest foot, what is the height of the cable above the ground at its lowest point? 6

7 10. Etension Eploration Each group will receive a quadratic equation to eamine. You will create a poster with the following information. If there are onl three group members, one person should do the tasks for group members #1 and #4. Group Member #1 Determine a = b = c = Does the parabola open up or down? b Determine the ais of smmetr: a Group Member # Enter the equation into the calculator Create a table verte Group Member #3 Graph the equation Identif the -intercepts Group Member #4 Solve the equation b factoring When ou re done, put a star or circle our Factored form Solutions X-intercepts 10. Etension Poster Summar Equation Factored Form Solutions -intercepts = = = 8 = = = = 4 = Wh is the factored form referred to as intercept form? What is the relationship between the solutions of the equation and the -intercepts? 7

8 Etension 10.: Graph Quadratic Functions in Intercept Form GRAPH OF INTERCEPT FORM: = a( p)( q) Step 1: Find the -intercepts b setting the factors equal to zero. Step : The Ais of Smmetr is halfwa between the roots. Step 3: Find and plot the verte. Opens Up if a-value is. Opens Down if a-value is. Eample 1: Graph = -( + 1)( 5) Step 1: Identif and plot the -intercepts Step : Find and draw the ais of smmetr: Step 3: Find and plot the verte. 8

9 Eample : Graph = ( + 1)( 3) Step 1: Identif and plot the -intercepts Step : Find and draw the ais of smmetr: Step 3: Find and plot the verte. Eample 3: = 8 Step 1: Rewrite the quadratic function in intercept form. Step : Identif and plot the -intercepts Step 3: Find and draw the ais of smmetr: Step 4: Find and plot the verte. 9

10 Section 10.3: Solve Quadratic Equations b Graphing NUMBER OF SOLUTIONS OF A QUADRATIC EQUATION Two Solutions One Solution No Solutions (Unfactorable) Eample 1: Solve + 4 = 5 b graphing. Step 1: Put equation into standard form. Step : Plug equation into calculator. Locate the intercepts ( = 0) and center the verte. Verte Step 3: Factor and solve the equation to check our roots. 10

11 Eample : Solve 4 = -4 b graphing. Step 1: Put equation into standard form. Step : Plug equation into calculator. Locate the intercepts ( = 0) and center the verte. Verte Step 3: Factor and solve the equation to check our roots. Eample 3: Solve + 8 = b graphing. Step 1: Put equation into standard form. Step : Plug equation into calculator. Locate the intercepts ( = 0) and center the verte. Verte Step 3: Factor and solve the equation to check our roots. 11

12 Section 10.4: Use Square Roots to solve Quadratic Equations Step 1: Get the squared term all b itself on one side of the equal sign. = (number) Step : To undo the square term, we have to do the inverse, which is to. If the (number) is positive, ou will have solution(s). (number) If the (number) is zero, ou will have solution(s). 0 If the (number) is negative, ou will have solution(s). Ø Step 3: Solve the remaining equations for the variable (if necessar). Eample 1: Solve the Equation. a. z 5 = 4 b. 5k = 9 c = 3 d. 6g + 1 = 19 e. 5( + 1) = 30 f. 3(m 4) = 1 1

13 Eample : During an ice hocke game, a remote-controlled blimp flies above the crowd and drops a numbered table-tennis ball. The number of the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air. Eample 3: You drop our glasses from a balcon 18 feet above our den on to a table that is 3 feet above the ground. How long are our glasses in the air? 13

14 Section 10.5 Solve Quadratic Equations b Completing the Square Use a bo to square the binomial. 1. ( + 3). ( + 4) 3. ( 7) What patterns do ou notice? Complete the Square: How can we reverse the procedure? Find the number that completes the square = ( + ) = ( - ) = ( + ) 4. + = ( - ) Let s Practice! Find the value of c such that each epression is a perfect square trinomial. Then write the epression as the square of a binomial c 6. r 4r + c 7. p 30p + c 14

15 Completing the Square Step 1: Put the epression into this form: + b + = (number) or b + = (number) Step : To fill in the blank, take half of b, square it and add it to both sides. + b + b b = (number) + or b + b = (number) + b Step 3: Factor the left side into a perfect square, simplif the right side. Step 4: Square root both sides to solve for. Eample 1: Solve the quadratic equation a. + 6 = -5 b. r 8r = 9 c. + 6 = 7 d. d 4 = 5d 15

16 Eample : Solve the equation b completing the square. a. 4m 16m + 8 = 0 b. 5s + 60s + 15 = 0 c = 0 d. k + 0k = 8 Eample 5: You are designing a herb garden with a uniform border of ornamental grass around it as shown below. Your design includes 1 square feet for the herb garden. Find the width of the grass border to the nearest inch. 16 ft 6 ft 16 6 ft

17 Section 10.5 Etension: Graph Quadratic Functions in Verte Form Sometimes quadratic functions are written: = a( h) + k This form of a quadratic is known as VERTEX FORM. Let s see if we can figure out wh. 1. = ( 3) + 4 Rewrite the given equation in standard form: = a + b + c. = ( + 1) Rewrite the given equation in standard form: = a + b + c Graph the equation b finding the ais of smmetr and creating a table. Graph the equation b finding the ais of smmetr and creating a table. verte verte 17

18 3. = -( + ) 3 Rewrite the given equation in standard form: = a + b + c 4. = ( + 3) + 1 Rewrite the given equation in standard form: = a + b + c Graph the equation b finding the ais of smmetr and creating a table. Graph the equation b finding the ais of smmetr and creating a table. verte verte Wh do ou think = a( h) + k is called verte form? 18

19 h Verte Form k a Verte: Ais of Smmetr: Up if: Down if: Eample 1: Graph a quadratic function in verte form = -( + ) + 3 Step 1: Identif the values of a, h, and k. Step : Identif and draw the ais of smmetr. Step 3: Find and plot the verte (h, k). Step 4: Find more points on the graph. Choose -values less than and greater than the verte. Verte 19

20 Eample : Write the function in verte form (complete the square) and graph the function. = Step 1: Write the function in verte form b completing the square. Step : Identif the values of a, h, and k. Step 3: Identif and draw the ais of smmetr. Step 4: Find and plot the verte (h, k). Step 5: Find more points on the graph. Choose -values less than and greater than the verte. Verte 0

21 Eample 3: Write the function in verte form (complete the square) and graph the function. a. = Step 1: Write the function in verte form b completing the square. Step : Identif the values of a, h, and k. Step 3: Identif and draw the ais of smmetr. Step 4: Find and plot the verte (h, k). Step 5: Find more points on the graph. Choose -values less than and greater than the verte. 1 Verte

22 Section 10.6: Solve Quadratic Equations b the Quadratic Formula The Quadratic Formula Used to solve quadratics equations in standard form. Standard Form: Quadratic Formula: b b 4ac a b c 0 a Eample 1: Use the quadratic formula to solve a. 5 = 3 b. + 7 = 9 c. 3 1 =

23 Eample : A crabbing net is thrown from a bridge, which is 35 feet above the water. If the net s initial velocit is 10 feet per second, how long will it take the net to hit the water? Eample 3: For the period , the number of book titles published b a small publishing compan can be modeled b the function = , where is the number of ears since In what ear did the compan publish 80 books? 3

24 Section 10.7: Interpret the Discriminant Discriminant Can determine how man solutions a quadratic equation has. b b 4ac a USING THE DISCRIMINANT OF a + b + c = 0 Value of the discriminant Number of solutions Graph of = a + b + c = 0 4

25 Eample 1: Find the discriminant and determine the number of solutions a. 3 = 0 b = -5 c. - 9 = 6 d. 3 + = 0 e = 0 f. + = 1 5

26 Section 10.8: Compare Linear, Eponential, and Quadratic Models Linear Function Eponential Function Quadratic Function = = = Eample 1: Use a graph to tell whether the ordered pairs represent a linear function, an eponential function, or a quadratic function. a. (-, 7), (-1, 1), (0, -1), (1, 1), (, 7) b. (-, 4), (-1,), (0, 1), (1, ½), (, ¼) 6

27 Differences and ratios in a table Linear equation Eponential function Quadratic Function Eample : Use differences or ratios to tell whether the table of values represents a linear function, an eponential function, or a quadratic function. Then write the equation for the function a b c

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