Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.

Transcription

1 Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to begin graphing quadratic functions. Ever quadratic has a U-shaped graph called a. The is ² Graph the parent function: Using the graph of the Quadratic Parent function, identif the following: * - or point on the parabola (Where the line changes direction) It is written as an ordered pair. * - the line that passes through the verte and divides the parabola into two smmetric parts * X = * are where the parabola crosses the -ais and represent the value when. Review changes in linear functions: What is the linear parent function? Changes in slope: If m > 1, If 0 < m < 1, Changes in -intercept: If b > 0, If b < 0, If m is negative, 1

2 1. Compare each linear equation to the graph of the linear parent function: =. A) 3 8 B) 1 3 C) 7 3 D) Write an equation of a line whose slope is steeper than the linear parent function and is translated up. E) If the line 8 is shifted up 6 units, but the slope remained unchanged, what will the new equation be? Changes in Quadratic Functions: Notice that a is the number is front of the ². a controls the direction in which the parabola opens. If a is positive then the parabola opens. The verte is a. If a is negative then the parabola opens. The verte is a. * parabola is reflected a also controls the width of the parabola. If > 1 then the parabola gets. This is called a. If 1 then the parabola gets. This is called a. For ² the -intercept is, so the point (, ) is on the parabola and is the. A change in c results in a vertical or shift from the parent function.. Comparing Graphs: A. B. Similarities: Differences: C. 1 D. Similarities: Differences: Unit 9 (Quadratic Functions) Notes

3 3. Graph the function = ² Verte: Minimum or Maimum Ais of smmetr: Root(s): Compare to parent function: 4. Graph the function = ½² Verte: Minimum or Maimum Ais of smmetr: Root(s): Compare to parent function: 5. Graph the function = -² Verte: Minimum or Maimum Ais of smmetr: Root(s): Compare to parent function: Unit 9 (Quadratic Functions) Notes 3

4 6. Graph the function = ²+ Verte: Minimum or Maimum Ais of smmetr: Root(s): Compare to parent function: 7. Graph the function = ²- 3 Verte: Minimum or Maimum Ais of smmetr: Root(s): Compare to parent function: 8. A Circle the functions that represent a reflection B For each, state if it is a stretch or a compression Unit 9 (Quadratic Functions) Notes 4

5 9. In the graph of the function 4, 10. When graphed, which function would which describes the shift in the verte appear to be shifted 5 units up from of the parabola if, in the function, the 4 the graph of f 5? is changed to 1? A 3 units up A f( ) 1 B 5 units up C 3 units down D 5 units down B C D f( ) 5 f( ) f( ) 1 Graphing a the parabola across the -ais and changes the parabola s. b moves the graph. c moves the graph. To find the ais of smmetr use the formula. To find the verte substitute in the function and solve for. Find the ais of smmetr and verte: 11. = a = b = c = What is the ais of smmetr? What is the verte? Minimum or Maimum 1. = a = b = c = What is the ais of smmetr? What is the verte? Minimum or Maimum 13. = a = b = c = What is the ais of smmetr? What is the verte? Minimum or Maimum Unit 9 (Quadratic Functions) Notes 5

6 To Graph a Quadratic Function: 1. Use the calculator and the table function. Graph the ordered pairs and sketch the u-shaped curve. The GRAPH of a Quadratic Function can be used to: Find the Solutions to a quadratic function. The solutions can also be called the roots, the zeroes, or the - intercepts. There could be one solution, two solutions or no solutions. Find the Verte and determine if it is the Maimum or Minimum point of the graph. Find the Equation for the Ais of Smmetr. 14. Graph the function = ² a = b = c = Ais of Smmetr: Verte Minimum or Maimum Roots: 15. Graph the function =- ² a = b = c = Ais of Smmetr: Verte Minimum or Maimum Roots: Unit 9 (Quadratic Functions) Notes 6

7 Remember that we can also solve a Quadratic Function b Factoring: 16. Given: 3, solve when = Domain And Range of Quadratic Functions Objective: Identif the domain and range of a quadratic function. Domain: Range: Arrows represent that the graph continues on, dots represent that the graph stops there. State the domain and range D = D = D = R = R = R = D = D = D = R = R = R = Unit 9 (Quadratic Functions) Notes 7

8 Quadratics and Other Functions Review: 4. Graph the function ² a = b = c = Ais of Smmetr: Verte Minimum or Maimum -intercepts: -intercept: Domain: Range: Compare to the parent function: 5. Graph the function ² a = b = c = Ais of Smmetr: Verte Minimum or Maimum -intercepts: -intercept: Domain: Range: Compare to the parent function: Unit 9 (Quadratic Functions) Notes 8

9 Solving a b c A is an equation that can be written in the standard form where a 0. Was to find the solution(s) or roots of a quadratic equation: 1. Factoring. Graphing/ Table 3. Quadratic Formula Ver Important:,,, ARE ALL THE SAME THING!!! To solve b graphing: 1. Put in standard from. Graph 3. Solution is where the parabola crosses the -ais Unit 9 (Quadratic Functions) Notes 9

10 To solve with a table: 1. Put in standard form. Enter the function into the calculator, pull up the table (ctl T) 3. The solutions are the values when =0 Factor and solve: Using the Quadratic Formula: 1. Put equation in the form: a b c 0. Identif the values of a, b and c 3. Plug values into the quadratic formula 4. Solve the epression Solve using the Quadratic Formula: b b ac a n 7n Unit 9 (Quadratic Functions) Notes 10

11 150. The arc of a baseball hit into the outfield can be modeled b the function where is the horizontal distance (in feet) that the baseball travels an is the height of the baseball. Does the baseball reach a height of 50 feet? If it does, about how far has it traveled horizontall from the hitter? *To find how man solutions a quadratic equation has ou can use the discriminant, b 4ac If the discriminant is: two solutions one solution no real solutions Use the discriminant to find out how man solutions each quadratic function will have: Unit 9 (Quadratic Functions) Notes 11

12 Identifing Functions LINEAR graphs as a straight line QUADRATIC graphs as a parabola EXPONENTIAL graphs as a curve with a sharp increase (GROWTH) or sharp decrease (DECAY) Sketch each parent function and one transformation. LINEAR QUADRATIC EXPONENTIAL m b a b c a b In each table, subtract the -values. What do ou notice? In b-c, subtract the -values again. What do ou notice? Subtract the -values in d. What do ou notice? a b c This process is called finite differences and it will allow ou to determine the degree of the function that created the table. If the differences are the same after one subtraction: If the differences are the same after two subtractions: Degree 1 Linear Degree Quadratic Unit 9 (Quadratic Functions) Notes 1

13 Eponential can be determined b the fact that the differences never get closer. The pattern for eponentials is that the are multiplied to get from one term to the net. Identif the tpe of function (Linear, Quadratic, Eponential, or None of these) from the table X Y /3 X Y X Y X Y Unit 9 (Quadratic Functions) Notes 13