Chapt 8 Quadratic Equations & Functions

8.1 Completing a Square

• Given: x 2 = u x = + √(u) or x = √(u) • E.g.

Given: x 2 = 3 x = √(3) or x = √(3)

Completing the Square

• Solve: x 2 • x 2 – 6x + 4 = 0 – 6x = -4 • How to make the left side a perfect square?

x 2 – 6x + 9 = -4 + 9 (x – 3) 2 = 5 x – 3 = √5 or x – 3 = -√5 x = 3 + √5 or x = 3 - √5 • Check: (3 + √5) 2 – 6(3 + √5) + 4 = 0 ?

9 + 6 √5 + 5 – 18 - 6√5 + 4 = 0 ?

(9 + 5 – 18 + 4) + (6√5 - 6√5 ) = 0 yes

Completing the Square

• Solve: 9x 2 – 6x – 4 = 0 • (9x 2 x 2 – 6x – 4)/9 = 0/9 – (6/9)x – 4/9 = 0 x 2 – (2/3)x = 4/9 x 2 – (2/3)x + (1/9) = 4/9 + (1/9) (x – 1/3) 2 = 5/9 x – 1/3 = √(5/9) or x – 1/3 = -√(5/9) x = 1/3 + √(5)/√(9) or x = 1/3 - √(5)/√(9) x = 1/3 + √(5)/3 or x = 1/3 - √(5)/3 x = (1 + √(5))/3 or x = (1 - √(5))/3

Your Turn

• Solve by completing the square: x 2 + 3x – 1 = 0 • x 2 + 3x = 1 x 2 + 3x + 9/4 = 1 + 9/4 (x + 3/2) 2 = 13/4 x + 3/2 = ± √(13/4) x = -3/2 ± √(13)/2 x = (-3/2 ± √(13)) / 2

• •

Your Turn

Solve by completing the square: 3x 2 + 6x + 1 = 0 3x 2 + 6x = -1 (3x 2 x 2 + 6x) / 3 = -1/3 + 2x = -1/3 x 2 + 2x + 1 = -1/3 + 1 (x + 1) 2 = 2/3 x + 1 = ±√(2/3) x = 1 ±√(2/3) x = 1 ±√(2/3) √(3)/√(3) x = 1 ±√(6) / 3 x = (3 ±√(6) ) / 3

Review

• Solve by completing the square.

(4x – 1) 2 = 15 • 16x 2 – 8x + 1 = 15 • 16x 2 • (16x 2 – 8x = 14 – 8x)/16 = 14/16 • x 2 – (1/2)x = 7/8 • x 2 – (1/2)x + 1/16 = 7/8 +1/16 • (x – ¼) 2 = 15/16 • x – ¼ = ±√(15/16) • x = ¼ + √(15/16) = ¼ + √(15)/4 = (1 + √(15)/4 • x = ¼ - √(15/16) = ¼ - √(15)/4 = (1 - √(15)/4

8.2 Quadratic Formula

• Given: ax 2 + bx + c = 0, where a > 0.

• x 2 x x 2 2 + (b/a)x + (c/a) = 0 + (b/a)x = -(c/a) + (b/a)x + b2/(4a (x + (b/(2a)) 2 2 ) = -(c/a) + b = -(c/a) + b 2 /(4a 2 ) 2 /(4a 2 ) (x + (b/(2a)) 2 = -(c/a)(4a 2 )/(4a 2 ) + b 2 /(4a 2 ) (x + (b/(2a)) 2 x + b/(2a) = ±√ ((b 2 x = (-b /(2a) = -((c/a)(4a 2 ) + b 2 ) / (4a 2 ) ±√ ((b 2 – 4ac)/(4a 2 )) – 4ac)/(2a) x = (-b ±√ (b2 – 4ac)) / (2a)

Quadratic Formula • Given: ax 2 + bx + c = 0, where a > 0.

• -b ± √(b 2 – 4ac) x = ---------------------- 2a • E.g., if 3x 2 – 2x – 4 = 0 a = 3; b = -2; c = -4 • 2 ±√(4 + 48) 2 ± √(52) 1 ± √(13) x = ---------------- = ------------ = -------------- 6 6 3

Application

• The number of fatal vehicle crashes has been found to be a function of a driver’s age. Younger and older driver’s tend to be involved in more fatal accidents, while those in the 30’s and 40’s tend to have the least such accidents. • The number of fatal crashes per 100 million miles, f(x), as a function of age, x, is given by f(x) = 0.013x

2 – 1.19x + 28.24

• What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven?

Fatal Crashes vs Age of Drivers

Age of US Drivers and Fatal Crashes

20.0

18.0

16.0

14.0

12.0

10.0

8.0

6.0

4.0

2.0

0.0

17.7

16 9.5

18 6.2

20 4.1

2.8

25 35 2.4

45

Age of Drivers

3.0

55 3.8

65 8.0

75 16.3

79

Solution

• f(x) = 0.013x

2 – 1.19x + 28.24

• 3 = 0.013x

2 – 1.19x + 28.24

• 0.013x

2 – 1.19x + 25.24 = 0 a = 0.013; b = -1.19; c = 25.24

• -(-1.19) ±√((-1.19) 2 – 4(0.013)(25.24)) x = ------------------------------------------- 2(0.013) • x ≈ ((1.19) ± √(0.104)) / (0.26) x ≈ ((1.19) ± 0.322) / (0.26) • x ≈ 33; x ≈ 58

Your Turn

• Solve the following using the quadratic formula 1. 4x 2 – 3x = -6 2. 3x 2 = 8x - 7

Discriminant

• Given: ax 2 + bx + c = 0, where a > 0.

• -b ± √(b 2 – 4ac) x = ---------------------- 2a • If (b 2 If (b 2 – 4ac) >= 0, x are real numbers – 4ac) < 0, x are imaginary numbers.

• (b 2 – 4ac) is called a

discriminant

. • Thus, – If (b 2 – 4ac) >= 0, solution set is real numbers.

– If (b 2 – 4ac) , 0, solution set is complex numbers.

Your Turn

• Compute the discriminant and determine the number and type of solutions.

1. x 2 + 6x = -9 2. x 2 + 2x + 9 = 0

8.3 Quadratic Function & Their Graphs

Quadratic function

f

(

x

) =

x

2 -

x

- 2

f(x) x

Characteristics of Quadratic Function Graph • f(x) = ax 2 + bx + c – Shape is a parabola – If a> 0, parabola opens upward – If a < 0, parabola opens downward – Vertex is the lowest point (when a > 0), and the highest point (when a < 0) – Axis of symmetry is the line through the vertex which divides the parabola into two mirror images.

To Sketch a Graph of Quadratic Function • Given: f(x) = a(x – h) 2 • Characteristics + k – If a > 0, parabola opens upward – Vertex is at (h, k) – If h > 0, graph is shifted to right by h; if h < 0, to the left – If k > 0, graph is shifted up by k; if k < 0, downward by k – Axis of symmetry: x = h – For x-intercepts, solve f(x) = 0

x -7 -6 -5 -4 -3 -2 -1 -0 1 2 3 4 5 6 7 f(x) -192 -154 -120 -90 -64 -42 -24 -10 0 6 8 6 0 -10 -24

Graph of f(x) = -2(x – 3)

2

+ 8

50

f(x) = -2(x - 3)^2 + 8

0 -50 -7 -6 -5 -4 -3 -2 -1 -0 1 2 3 4 5 6 7 8 9 10 11 -100 -150 -200 -250

x Excel

Graph of f(x) = -2(x – 3)

2

+ 8

• Graph the function: f(x) = -2(x – 3) 2 • Solution: – Parabola opens downward (a = -2) – Vertex: (3, 8) – X-intercepts: -2(x – 3) 2 + 8 = 0 (x – 3) 2 = -8/-2 x – 3 = ± √(4) x = 5; x = 1 – y-intercept f(0) = -2(0-3) 2 = -2(9) + 8 + 8 = -10 + 8

x -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Graph of f(x) = (x + 3)

2

+ 1

10.0

17.0

26.0

37.0

50.0

65.0

82.0

101.0

f(x) 17.0

10.0

5.0

2.0

1.0

2.0

5.0

120.0

100.0

80.0

60.0

40.0

20.0

0.0

f(x) = (x + 3)^2 + 1

-7 -6 -5 -4 -3 -2 -1 0 1 2 3

x

4 5 6 7

Graph of f(x) = (x + 3)

2

+ 1

• Graph the function: f(x) = (x + 3) 2 + 1 • Solution: – Parabola opens upward (a = 1) – Vertex: (-3, 1) – X-intercepts: (x + 3) 2 + 1 = 0 (x + 3) 2 x – 3 = ± = -1

i

x = 3 +

i

; x = 3 -

i

(This means no x-intercepts) – Y-intercept f(0) = (3) 2 + 1 = 10

Graphing f(x) = ax

2

+ bx + c

• A • Solution: f(x) = a(x 2 + (b/a)x) + c = a(x 2 + (b/a)x + (b 2 /4a 2 )) + c – a(b 2 /4a 2 ) = a(x + (b/2a)) 2 + c – b 2 /(4a) • Comparing to f(x) = a(x – h) 2 h = -b/(2a); k = c – b 2 /(4a) + k, • f(x) = a(x – (-b/(2a)) 2 + (c – b 2 /(4a))

Graphing f(x) = 2x

2

+ 4x - 3

• a = 2; b = 4; c = -3 • f(x) = a(x – (-b/(2a)) 2 = 2(x – (-4/(4)) 2 = 2(x + 1) 2 – 5 + (c – b 2 /(4a)) + (-3 – 16/8) • Opens upward • Vertex: (-1, -5) • X-intercept: -1 √(10)/2, -1 + √(10)/2 • Y-intercept: f(0) = -3

Graphing f(x) = 2x

2

+ 4x - 3

f(x) = 2x^2 + 4x - 3

140.0

120.0

100.0

80.0

60.0

40.0

20.0

0.0

-20.0

-7 -6 -5 -4 -3 -2 -1 0 1

x

2 3 4 5 6 7