4-1 Graphing Quadratic Functions

Transcription

1 4-1 Graphing Quadratic Functions Quadratic Function in standard form: f() a b c The graph of a quadratic function is a. y intercept Ais of symmetry -coordinate of verte coordinate of verte 1) f ( ) 4 a= b= c= Find (a) y-intercept, (b) ais of symmetry, (c) the verte coordinates, and (d) make a table with numbers above and below the verte. (a) (b) (c) (d) ² f() Maimum and Minimum: If a 0, the parabola opens and the verte is a. If a 0, the parabola opens and the verte is a. ) Will the following quadratic functions open up or down? a) b) f( ) 4 therefore the verte is a min or ma? f( ) therefore the verte is a min or ma?

2 To graph a quadratic function, follow these steps: a. Find the y-intercept b. Find the ais of symmetry c. Find the verte d. Ask yourself, will the parabola open up or down? o Is the verte a ma or min? e. Make a table with numbers above and below the verte. 3) f ( ) 8 9 (a) (b) (c) a= b= c= (d) (e) ² f() Domain: Range: 4) An object is fired straight up from the top of a 00-foot tower at a velocity of 80 feet per second. The height h(t) of the object t seconds after firing is given by h(t) = -16t² + 80t Find the maimum height reached by the object and time that the height is reached.

3 4- Solving Quadratic Equations by Graphing Quadratic Equation: The solutions of a quadratic equation are called. One method to find the is to find the. The of the function are actually the of its graph. You can find the of some quadratic equations by. 1) Determine the number of solution. a) b) c) Remember the five steps to graphing a quadratic these are on your notes from 4-1!! Identify: a= b= c= (a) (b) (c) (d) (e)

4 ) Graph: f( ) 4 5 a= b= c= (a) y-int: (b) ais of sym: (c) verte: (d) opens: (e) f() Zeros: Domain: Range: 4) Graph: g( ) 6 5 a= b= c= (a) y-int: (b) ais of sym: (c) verte: (d) opens: (e) g() Zeros: Domain: Range:

5 4-3 Solving Quadratic Equations by Factoring Consider: ab 0 Either a 0 and b # or b 0 and a # Zero Product Property Eample: If ( 5)( 7) 0 then or. Let s Try! Solve by factoring

6 We can work backwards also! Given the roots, write the quadratic equation , , 5 15., , ,4 3 Real World: 18. The entrance to an office building is an arch in the shape of a parabola whose verte is the height of the arch. The height of the arch is given by h 9, where is the horizontal distance from the center of the arch. Both h and are measured in feet. How wide is the arch at the ground level?

7 4-4 Comple Numbers and Roots Imaginary Numbers: 1 i and i 1 Simplify. Epress each number in terms ofi. 1) 49 ) 8 3) 3 4 4) ) 4i 5i 6) 3i i 7) 1 8) Evaluating Powers of i. 1 i i i i i i i 6 i i 3 i 1 1 i i i i i i i i i i i i i i 1 1 9) 6 i 10) 6 i 11) 45 i 1) 35 i 13) 40 i 14) 63 i

8 Equations with Imaginary solutions: Solve each equation. 15) ) ) ) Comple Numbers: a bi Equating Two Comple Numbers: Find the values of and y that make the equations true. 19) 6i 8 (0 y) i 0) 8 (6 y) i 5 i 6 1) 4 10i (4 y) i Adding and Subtracting Comple Numbers: Simplify and write result in a bi form. ) 3 5i 6i 3) i 3 5i 4) 4 3i 4 3i 5) 5 i 3i Multiplying Comple Numbers: Simplify and write result in a bi form. 6) i 3 5i 7) 4 4i 6 i 8) 3 i 3 i

9 Comple Conjugates: Find each comple conjugate (other solution). 9) 9 i 30) i 3 31) 8i 3) 5i 7 Dividing Comple Numbers: Simplify and write result in a bi form. 33) 3 10 i 5i 34) 5i 3 i 35) 3 i i REAL WORLD: Electricity In an AC circuit, the voltage E, current I and impedance Z are related by the formula E = I Z. Find the voltage in a circuit with current 1 + 4i amps and impedance 3 6i ohms.

10 Square Root Property 4-5 Completing the Square (Another way to solve a quadratic equation!) 1) ) ) y 1y ) w 10w 5 3 5) ) What do you do if the quadratic isn t a perfect square??? MAKE THEM PERFECT SQUARES hence the term Completing the Square!!! To complete the square for a quadratic epression of the form b c b c Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 7) 8 c 8) 1 c 9) 5 c 10) c 1

11 To solve a quadratic equation by completing the square, follow these steps: move the constant to the right side 0 36 leave a space for c and put a space on the right side to balance find c and write it in the spaces on the left and right write the left side as a perfect square and simplify the right side take the square root of both sides 10 8 solve, 18 Solve each equation by completing the square. 11) ) ) ) ) Comple Zeros of Quadratic Equations: Find the zeros of each function by completing the square. 16) ) )

12 4-6 The Quadratic Formula and Discriminant Quadratic Formula: 1) Solve 8 33 by using the quadratic formula. a= b= c= = = 4 The solution(s) is(are): ) Solve by the quadratic formula. a= b= c= = The solution(s) is(are):

13 3) 4) 5 The solution(s) is(are): The solution(s) is(are): In b Is the discriminant a 5) Find the values of the discriminant. Describe. a. b b 4ac 0 ( pos) real rational b 4 ac is a perfect square b 4ac 0 ( pos) real irrational b 4 ac is not a perfect square b 4ac 0 1 real rational c d b 4ac 0 ( neg ) comple

14 4-7 Transformations of Quadratic Graphs Verte Form: The parent graph: Verte: (, ) k: moves parabola or h: moves parabola or a: makes parabola or **a small then **a big then Write each function in verte form. Identify the verte. 1) f( ) 16 1 ) f( ) 4 3) f( ) 1 4 4) f( )

15 5) Match the functions listed below to the correct graphs. (a) y ( 3) 1 (b) y ( 1) 3 (c) 1 ( 1) 3 y (d) y ( 3) 1 (e) y ( 1) 3 (f) 1 ( 1) 3 y Graph each function: 6) y 3 1 7) y 1 8) y 4 4 9) y 3 4