1. Chapt 8 Quadratic Equations & Functions

2. 8.1 Completing a Square Given: x2 = u x = +√(u) or x = -√(u)
E.g. Given: x2 = 3 x = √(3) or x = -√(3)

3. Completing the Square Solve: x2 + 6x + 4 = 0 x2 + 6x = -4
How to make the left side a perfect square?
x2 + 6x + 9 =
(x + 3)2 = 5
x + 3 = √ or x + 3 = -√5
x = -3 + √ or x = -3 - √5
Check: (-3 + √5)2 + 6(-3 +√5) + 4 = 0 ? 9 - 6√ √5 + 4 = 0 ? (9 + 5 – ) + (-6√5 + 6√5 ) = 0 yes

4. Completing the Square Solve: 9x2 – 6x – 4 = 0 9x2 – 6x = 4
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)/ or x = 1/3 - √(5)/3 x = (1 + √(5))/ or x = (1 - √(5))/3

5. Your Turn Solve by completing the square: x2 + 3x – 1 = 0 x2 + 3x = 1

6. Your Turn Solve by completing the square: 3x2 + 6x + 1 = 0
(3x2 + 6x) / 3 = -1/3 x2 + 2x = -1/3
x2 + 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 ±√((2)(3 ))/ 3
x = -1 ±√(6) / 3 x = (-3 ±√(6) ) / 3

7. Review Solve by completing the square. (4x – 1)2 = 15
16x2 – 8x + 1 = 15
16x2 – 8x = 14
(16x2 – 8x)/16 = 14/16
x2 – (1/2)x = 7/8
x2 – (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. 8.2 Quadratic Formula Given: ax2 + bx + c = 0, where a > 0.
(ax2 + bx)a = -c/a
x2 + (b/a)x = -c/a
x2 + (b/a)x + (b/2a)2 = -c/a + (b/2a)2
(x + (b/(2a))2 = -c/a + b2/(4a2)
(x + (b/(2a))2 = -(c/a)(4a)/(4a) + b2/(4a2)
(x + (b/(2a))2 = (-4ac) + b2) / (4a2)
x + b/(2a) = ±√ ((b2 – 4ac)/(4a2)) x = (-b /(2a) ±√ ((b2 – 4ac)/(2a) x = (-b ±√ (b2 – 4ac)) / (2a)

9. Quadratic Formula Given: ax2 + bx + c = 0, where a > 0.
-b ± √(b2 – 4ac) x = a
E.g., if x2 – 2x – 4 = a = 3; b = -2; c = -4
2 ±√(4 + 48) 2 ± √(52) 1 ± √(13) x = = =

10. 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) = 0.013x2 – 1.19x
What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven?

11. Fatal Crashes vs Age of Drivers

12. Solution f(x) = 0.013x2 – 1.19x + 28.24 3 = 0.013x2 – 1.19x + 28.24
0.013x2 – 1.19x = 0 a = 0.013; b = -1.19; c = 25.24
-(-1.19) ±√((-1.19)2 – 4(0.013)(25.24)) x = (0.013)
x ≈ ((1.19) ± √(0.104)) / (0.26) x ≈ ((1.19) ± 0.322) / (0.26)
x ≈ 33; x ≈ 58

13. Your Turn Solve the following using the quadratic formula 4x2 – 3x = 6
a = 4; b = -3; c = -6
3 ±√((9 – 4(4)(-6)) ±√(105) x = = (2)(4)
3x2 = 8x + 7
3x2 - 8x - 7 = 0
a = 3; b = -8; c = -7
8 ±√(64 – 4(3)(-7)) ±√(148) ±√(4 ·37)
x = = =
4 ±√(37) = 3

14. Discriminant Given: ax2 + bx + c = 0, where a > 0.
-b ± √(b2 – 4ac) x = a
If (b2 – 4ac) >= 0, x are real numbers If (b2 – 4ac) < 0, x are imaginary numbers.
(b2 – 4ac) is called a discriminant.
Thus,
If (b2 – 4ac) >= 0, solution set is real numbers.
If (b2 – 4ac) , 0, solution set is complex numbers.

15. Your Turn
Compute the discriminant and determine the number and type of solutions.
x2 + 6x = -9
x2 + 2x + 9 = 0

16. 8.3 Quadratic Function & Their Graphs

17. Quadratic function f(x) = x2 - x - 2

18. Characteristics of Quadratic Function Graph
f(x) = ax2 + 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.

19. To Sketch a Graph of Quadratic Function
Given: f(x) = a(x – h)2 + k
Characteristics
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

20. Graph of f(x) = -2(x – 3)2 + 8 Excel x f(x) -7 -192 -6 -154 -5 -120 -4
-90 -3
-64 -2
-42 -1
-24 -0
-10 1 2 6 3 8 4 5 7
Excel

21. Graph of f(x) = -2(x – 3)2 + 8
Graph the function: f(x) = -2(x – 3)2 + 8
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(9) = -10

22. Graph of f(x) = (x + 3)2 + 1 x f(x) -7 17.0 -6 10.0 -5 5.0 -4 2.0 -3
1.0 -2 -1 1 2
26.0 3
37.0 4
50.0 5
65.0 6
82.0 7
101.0

23. 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 = -1 x – 3 = ± i x = 3 + i; x = 3 - i (This means no x-intercepts)
Y-intercept f(0) = (3)2 + 1 = 10

24. Graphing f(x) = ax2 + bx + c
Solution: f(x) = a(x2 + (b/a)x) + c = a(x2 + (b/a)x + (b2/4a2)) + c – a(b2/4a2) = a(x + (b/2a))2 + c – b2/(4a)
Comparing to f(x) = a(x – h)2 + k, h = -b/(2a); k = c – b2/(4a)
f(x) = a(x – (-b/(2a))2 + (c – b2/(4a))

25. Graphing f(x) = 2x2 + 4x - 3 a = 2; b = 4; c = -3
f(x) = a(x – (-b/(2a))2 + (c – b2/(4a)) = 2(x – (-4/(4))2 + (-3 – 16/8) = 2(x + 1)2 – 5
Opens upward
Vertex: (-1, -5)
X-intercept: -1 - √(10)/2, -1 + √(10)/2
Y-intercept: f(0) = -3

26. Graphing f(x) = 2x2 + 4x - 3