1. College Algebra
Acosta/Karwowski

2. Quadratic Functions and circles
Unit 3
Quadratic Functions and circles

3. Solving quadratic equations
Chapter 4 – section 1

4. A quadratic is any equation with a 2nd degree term and no higher degree terms
In general form ax2 + bx + c = 0
f(x) = ax2 + bx + c is the general form of a quadratic function

5. Finding f(x)=a : solving
In general the inverse of a power is its root
you can cancel square powers with square roots if you account for restrictions on the root function
2 𝑥 2 ≠𝑥 for every x
2 𝑥 2 =|𝑥|
thus: if 𝑥 2 =25 𝑖𝑡 𝑓𝑜𝑙𝑙𝑜𝑤𝑠 𝑡ℎ𝑎𝑡 𝑥=±5

6. Reminders : working with square roots
If the square root is irrational – leave the radical form of the answer unless ASKED to round the answer - This is called the exact form of the number
ex: ans: 4
ans: 21
Simplify all radicals
𝑎𝑛𝑠: 21
𝑎𝑛𝑠: 4 5
Square root of a negative is an IMAGINARY number
−4 =2𝑖 NOT - no solution or dne
−12 =2𝑖 3

7. Examples: solve using square root
5x2 – 3 = f(x) = 5x2 – 3 find f(x) = 77
3x2 + 7 = 25
(x – 4)2 = 16
24 + (x – 3)2 = 15
(comment on standard form)
x2 +8x = 15

8. Draw back to inversing
If x does not appear in the problem only once (in other words the function is a combination of linear and square) then using square root to isolate the x becomes problematic since:
A. You cannot combine unlike terms
3x2 + 5x
B. You cannot do the square root on an expression that contains addition outside of ( )
𝑥 −5 2 =𝑥 −5
𝑥 2 −4𝑥 ≠ 𝑥 −2

9. Using factors to solve quadratics
Many quadratics will factor
ex: x2 – 7x + 12 =
3x x + 20 =
9x2 – 121 =
15x2 + 35x – 90 =

10. Mathematical fact
if ab = 0 either a= 0 or b = 0

11. Thus: zero product rule is born
Given a quadratic equation, you can SOMETIMES isolate the x by
1. Making the equation = 0
2. Factoring the quadratic
3. Splitting the one equation into 2 and Solving each equation

12. Examples x2 + 6x – 16 = 0 x2 – 7x = -12 (x - 3)(x – 8) = 66
Next hurdle: x2 -10x + 18 = 0 ??????

13. Bridging the gap
When a problem cannot be done by factoring because they do not have integral factors and cannot be solve by square root because there is an x – term in the problem then:
These problems can be re written so that the x only appears in the equation once.
The process is called completing the square

14. Completing the square
Concept - change a quadratic so that it is a square quadratic
ie: so it factors (x +a)(x + a) which is (x + a)2
Given the first 2 terms you can determine the 3rd term
Fill in the blanks x2 + 8x + ___ = (x + ___)2
x2 – 10x + ___ =(x + ___)2
x2 + 5x + ____=(x + ___)2
Filling in these blanks is called completing the square

15. Completing the square to solve equations
4x2 – 40x = 0
Prepare for completing the square by dividing by 4 and moving the 20 ( which will be a 5)
x2 – 10x = 5
Completing the square CHANGES the number so add the same amount to BOTH sides of the equation
x2 – 10x + 25 =
(x – 5)2 = x = 5± 30

16. Examples : solve the following
3x2 – 24x + 6 = 0
x2 – 7x – 3 = 0
3x2 – 11x – 4 = 0
x2 + 5x + 15 =0

17. Interesting to note All quadratics can be solved
All quadratics are transformations of f(x)= x2

18. Guaranteed test question – 10 pts
Mathematics finds the formulas –
You find a formula by solving without using any known numbers
Solve ax2 + bx + c = for x thereby deriving a formula for x

19. Deriving the quadratic formula Solve ax2 + bx + c = 0 by completing the square
Divide by a :
Move constant to the right
Divide middle coefficient by 2 /square/ add
Get a common denominator and combine fraction
Write as a square
Isolate the x

20. Quadratic functions
Chapter 4 – Section 2

21. General overview
All quadratic functions are transformations of the parent f(x) = x2
Significant elements of quadratic function:
y-intercept
x-intercept (root/zero equation)
maximum/minimum (vertex)
concavity
line of symmetry
increasing/decreasing intervals
domain/range

22. Quadratic functions- useful forms
f(x) = ax2 + bx + c = y is a quadratic function
in general form
f(x) = a(x – h)2 + k = y can be the SAME quadratic function - in standard form
f(x) = a(x – x1)(x – x2) can be the same function also – in factored form

23. Changing forms- can be done but is not necessary
To general form from factored or standard – simplify
ex: g(x) = 3(x – 2)(x – 2/3)
k(x) = 5(x – 7)2 - 9
To factored form from general – factor the polynomial if possible
ex: m(x) = 2x2 – 3x – 20
To standard form from general ( a variation on completing square)
ex: f(x) = 2x2 – 12x + 10

24. Examples: find x and y intercepts
f(x) = x2 – 12x +32
g(x) = (x + 2)(x – 9)
k(x) = 3(x – 6)

25. An important correlation
solutions to the zero equation – x-intercepts – factors are significantly related
x = 5 is a solution to the zero equation if and only if (5,0) is an x – intercept and (x – 5) is a factor of the quadratic
If x = ……
If x = 4 – 9i ……

26. Concavity- value of a Oriented up(a >0)/ down a<0
Width (rate of change) - as |a | increases the parabola stretches vertically and becimes narrower
ex: f(x) = 3x2 – 5x + 12
g(x) = 5 – 2x – 3x2
k(x) = (x + 2)2 – 9
m(x) = (2 – x)(x + 5)
Related concept – max/min and range

27. Find vertex- g(x) = 3(x + 5)2 + 3 Related concepts - line of symmetry
range
intervals of inc/dec

28. Finding vertex etc. f(x) = -2x2 + 4x – 7
method one – easiest – uses information from quadratic formula
Method two – change to vertex form – (not recommended)
Related concepts - line of symmetry
range
intervals of inc/dec

29. Find vertex k(x) = (x – 9)(x + 3)
typical method: put into general form
Food for thought - line of symmetry
(midpoint is found by average)
Related concepts - line of symmetry
range
intervals of inc/dec

30. Sketching a graph for a quadratic by hand
Find vertex and at least one other point- use symmetry or table of solutions to find a third point
Ex: f(x) = 2x2 – 3x – 20
Ex: g(x) = 3(x – 5)(x + 3)
Ex: k(x) = -(x – 5)2 – 2

31. significance
A ball is thrown into the air has a height given by the function h(x) = -16x2 + 34x wherw x is the time that has elapsed since throwing the ball
Find the y intercept and interpret its meaning?
When will the ball hit the ground?
Where will the ball be in 1.2 seconds?
When will the ball be 17 feet above ground?
How high did the ball go?
Find h(3) and explain its significance..

32. Writing equations – word problems
To write an equation for a quadratic you need either
3 random points
or the vertex and one other solution point
Or factors and one other point
or a formula which has already been derived by someone else.

33. Physics formula the quadratic function h(x) = ax2 + bx + c
has a physics application where
a = acceleration of an object after it is released (usually
this is gravitional acceleration)
b = its initial velocity (force with which it is released)
c = the original location of the object
Write a function for an object with gravitational acceleration (known to be -16 ft/sec2) an initial velocity force of 12 ft/sec and an initial location of 29 feet.

34. Banking formula- compound interest
P(r) = p0 (1 + r)t
where p0 is original amount in bank
t is a set amount of time
r is the rate of interest
note: t must be the same units as r
if r is 6% per month then t = 7 is 7 months
Ex. Mark invests $4500 for 3 years. Write a function for his account as a function of r.

35. Given 3 point
(2,13) (-3, 38) ( 1,6)

36. Given vertex and one point
(2,4) is the vertex and the graph goes through the point (3,10)

37. examples
Given the points (5,0) (-2,0) and the point (3,-2) write an equation – you could do a system of equations since this is 3 points
Or -
a(x – 5) (x + 2)= 0 must be true
Thus a(x -5)(x +2) = y is the basis for the equation and
a(3 – 5)(3 +2) = -2
so -10a = -2 and a = 1/5
answer g(x) = 1/5 (x – 5)(x + 2)

38. Examples Irrational and imaginary solutions come in pairs
given intercept , 0 You know there is another intercept – 2− 3 , 0
You have factors
[x -(2+ 3 )] [x- (2− 3 )]
You can simplify the quadratic and find the stretch factor given a point (-4,3)

39. example Given solution - x = 3i and point (3, - 2)
You know another solution is x = - 3i
You have factors
You simplify and find the stretch factor

40. Quadratic inequalities in one variable
Chapter 4 – Section 3

41. What we already know
The solutions of inequalities in one variable are intervals on the number line
The solutions to inequalities are directly related to the solution of equations
There are 2 solutions to quadratic equations.

42. What the graph of the quadratic function tells us
find f(x) = 0
find f(x)>0
find f(x)<0
So quadratic inequalities work like absolute value inequalities

43. Examples find x2 – 9x >0 Find 2x2 + 7x – 30 < 0

44. circles Distance formula 𝑎 2 + 𝑏 2 = 𝑐 2
𝑥 1 − 𝑥 𝑦 1 − 𝑦 = 𝑑 2
A circle is the set of all points that are the same distance from a given point

45. standard form
Given (h,k) is the center and r is the radius the distance formula gives us
(x – h)2 + (y – k)2 = r2
This textbook calls this standard form for the circle equation
With (h, k) = (0,0) you have a circle centered on the origin so standard form is a transformation of 𝑥 2 + 𝑦 2 =1 (called the unit circle)

46. Graphing circles
(x – 5)2 + (y + 2)2 = 16

47. Writing the equation (x – h)2 + (y – k)2 = r2
Given center and radius simply fill in the blanks
Example:
A circle with radius 5 and center at (-2, 5)
Given center and a point - find radius and fill in blanks
A circle with center at (4,8) that goes through (7, 12)

48. General form Standard form of circle (transformation form)
Get standard form from general form.
𝑎𝑥 2 +𝑏𝑥+𝑎 𝑦 2 +𝑐𝑦=𝑟
3 𝑥 2 +12𝑥+3 𝑦 2 −30𝑦=6

49. Quadratic equations: factoring /square root/completing square/quadratic formula
Derive quadratic formula
Quadratic function: standard form a(x-h)2+k = y
general form ax2 +bx + c =y
factored form a(x- x1)(x-x2)
Concavity
Find x-int(roots) y-intercepts
Find vertex (h,k) x = -b/2a
Domain range
intervals of increase, or decrease
Write quadratic equations
Quadratic inequalities
Circles – standard form
general form