Flashcards in BASIC QUADRATIC FACTS Deck (92)
1
Q

What is the equation of the axis of symmetry?

A

x = -b / 2a

1
Q

What are the roots of a function? (other names?)

A

the zeros, the x-intercepts, the solutions, where the parabola crosses

1
Q

Projectile motion:

How do you find at what time it is at its peak?

h(t)= -16t2 + 44t + 95

A

the time at the peak is

t=-b/2a

plug in b and a to find t

2
Q

f(x)= 3x - 9

What is a, b and c?

A

a = 0 , b = 3, c = 9

2
Q

What are the zeros of a function? (other names?)

A

the roots, the x intercepts, the solutions, where the parabola crosses

2
Q

How do you know if a parabola opens upward (smile) or down (frown)?

A

if “a” is pos it opens up? if a is negative, it opens down.

2
Q

If there is no constant term, what is an easy way to factor?

A

just factor out the x

and set the factors to zero (one solution is zero)

3
Q

What is the zero product property?

A

(this)(that)=0

is true when

(this)=0 and when (that)=0

This is why we set factors = 0 when we solve quadratics.

4
Q

f(x)= 3x2-4x+5 What is a, b and c ?

A

a = 3, b = -4, c = 5

5
Q

what completes the square and what is the factored form? x2 + 5x + ____

A

25/4 completes it

and

(x + 5/2) 2

is factored form

5
Q

How do you solve by isolating the x2 ?

A

When there is no linear term (no x term), simply act like you’re solving for x, but solve for x2 and in the last step, take square root of both sides ( + / - )

6
Q

f(x) = ax2 + bx + c

A

ax2

7
Q

thinking of the quadratic formula, why are there 2 real roots when the discriminant is positive?

A

with the square root of a positive, you will have to +- that in the top, giving 2 different real solutions.

8
Q

Explain what a parabola with one real root (real zeros) looks like.

A

just touches x axis at one spot, the root (the zero), it doesn’t cross it, it bounces off it

10
Q

f(x)= 3x2-6x+3 What is the constant term?

A

3

it is the y intercept

10
Q

f(x)= 2x2-6x+3

What is the coefficient of the linear term?

A

-6

this is also known as

“b”

10
Q

what is the quadratic term’s exponent on the x ?

A

2

11
Q

Projectile motion:

If you are answering a “when” question, what are you substituting in for?

A

you will substitute in for h(t) and then solve for the t

12
Q

why is it called a constant term

A

alone it would be a constant function with a constant y-value (height)

12
Q

Describe what an axis of symmetry is?

A

it is where the parabola can fold onto itself

13
Q

Does “a” have to be equal to 1 when doing quadratic formula?

A

NO

14
Q

Explain what a parabola with two real roots (real zeros) looks like.

A

crosses x axis at those roots, at those x values

15
Q

Does “a” have to be equal to 1 when completing a square?

A

YES

17
Q

What are solutions to

(x+8)(x-3) = 0

A

x = { -8, 3}

this is not a point on the graph, it is a set of two numbers, or two x values.

18
Q

What do we call b2-4ac

A

discriminant

19
Q

what completes the square and what is the factored form? x2 + 6x + ____

A

9 completes it and (x+3)2

19
Q

If the discriminant is positive, how many times does the parabola hit the x axis?

A

2

20
Q

When can you just isolate the x2 and root both sides?

A

when there are no linear terms (only quad and const)

21
Q

what is difference between (3, -2) and {3, -2}

A

the first is a point on a plane (over 3, down 2). The second is a set containing two values, 3 and -2, not a point.

22
Q

f(x)= -4x2 + 8

What is a, b and c ?

A

a = -4, b = 0, c = 8

22
Q

If the discriminant is zero, what do you know about roots?

A

there is

one real double root

24
Q

Why is it called a “quadratic?”

A

From the latin “quadrus” which means “square”

24
Q

What does the discriminant tell us about?

A

whether they are real or not

25
Q

Projectile motion: How do you find the time when it is 10 ft high?

h(t)= -16t2 + 48t +350

A

put 10 where h(t) is. Solve quadratic for t

27
Q

h(t)= -16t2 + 55t - 60

what is the “-16”

A

that is the pull of gravity, actually, half of gravity. Gravity pulls at 32ft/sec2 at sea level.

29
Q

What are the five ways to solve a quadratic?

A

isolate x2, factor, complete square, quadratic formula, graph and look for x-intercepts

29
Q

How do you solve by completing the square?

A

get all stuff on one side. Make sure a=1, take half of b and square it, add to both sides.

30
Q

f(x)= 3x2-6x+5

What is the coefficient of the constant term?

A

5

this is also known as

“c”

32
Q

What are the x intercepts of a function? (other names?)

A

the roots, the zeros, the solutions

33
Q

how would you solve

(3x-2)(x-5)=0

(what are solutions?)

A

set factors = 0 and solve:

3x-2=0 and solve x-5=0

zero product property solutions x = { 5, 2/3}

34
Q

what is the constant term’s exponent on the x?

A

0

35
Q

Projectile motion:

If you are answering a “what height” question what are you solving for?

A

h(t).

You are trying to find a height so you plug in a t

36
Q

h(t)= -16t2 + 77t +45 what is the initial velocity?

A

77 ft/sec

38
Q

f(x)= 5 - 2x +3x2 What is a, b and c ?

A

a = 3, b = -2, c = 5

38
Q

f(x) = ax2 + bx + c

What is the linear term?

A

bx

40
Q

Projectile motion: How do you find the height after 10 seconds?

h(t)= -16t2 + 32t +100

A

plug 10 into function

(where the t’s are)

and you’ll find h(t)

41
Q

Projectile motion: What are the only two things you are trying to solve?

h(t)= -16t2 + 32t +100

A

Either trying to find

height, h(t),

or time, t

42
Q

h(t)= -16t2 + 25t +100 what is the “25”

A

that is the initial velocity, or starting speed. 25 ft/sec

43
Q

Projectile motion:

If you are answering a “what height” question what are you substituting in for?

A

you substitute a time in for t and then find the h(t)

44
Q

Explain what a parabola with no real roots (real zeros) looks like.

A

doesn’t touch x axis

45
Q

h(t)= -16t2 + 12t +10 what is the “10”

A

that is the initial height, or starting height, y intercept. 10 ft.

47
Q

What is the discriminant?

A

b2-4ac

(notice there is no sq root on it!!)

47
Q

How do you find the max or min value of a quadratic?

A

it is at the vertex. Plug -b/2a into the function to get y. that y value is the maximum or minimum value

48
Q

Projectile motion:

how do you find the maximum height?

h(t)= -16t2 + 32t +100

A

find -b/2a and plug into function to find h(t).

49
Q

thinking of the quadratic formula, why are there no real roots when you have a negative discriminant?

A

because you have to take the square root of a negative, which is an imaginary number. you end with an imaginary..hence.. Imaginary roots

51
Q

If the discriminant is negative, what do you know about roots?

A

there are no real roots,

2 imaginary roots,

it doesn’t touch x axis

51
Q

Projectile motion:

How do you find the time when it is 40 ft high?

h(t)= -16t2 + 32t +100

A

put 40 where h(t) is.

52
Q

h(t)= -16t2 + 32t +600

what is the y intercept?

A

600, It is the initial height

53
Q

why is it called a discriminant?

A

because it discriminates between which type of roots you will have

54
Q

Projectile motion:

how do you find the time it hits the ground?

h(t)= -16t2 + 88t +100

A

this is when the height is zero, when h(t)=0.

so put a zero where h(t) is and solve the quadratic for t

55
Q

h(t)= -16t2 + 66t +100 what is the initial height?

A

100 ft

56
Q

How do you know when it is a quadratic?

A

when there is an x2 term (quadrus:square)

58
Q

f(x)= 3x2-6x+3 What is the linear term?

A

-6x

59
Q

f(x) = ax2 + bx + c

What is the constant term?

A

c

(it is the y intercept)

61
Q

what completes the square and what is the factored form? x2 - 8x + ____

A

16 completes it and (x-4)2

62
Q

How do you solve by factoring?

A

get stuff all on one side, factor, set each factor = 0, solve each.

64
Q

why are roots called “zeros”

A

because it is where the y value is zero, (the height of function is zero)

66
Q

How do you know if a parabola has a max or minimum by looking at the function?

A

if “ a” is pos, it has a min. if a is negative, it has a max

67
Q

When we graph a parabola to solve a quadratic, what do we look for?

A

look for the x intercepts, these are the roots. The zeros, the solutions?

68
Q

what completes the square and what is its factored form? x2 + b/a x + ____

A

b2/4a2 completes it

and

(x+ b/2a)2

69
Q

what is a coefficient?

A

the number in front of a variable or variables

70
Q

what is the linear term’s exponent on the x ?

A

1

71
Q

If the discriminant is positive, what do you know about roots?

A

there are 2 real roots, and the parabola crosses x axis at these roots

72
Q

Geometrically, how is the shape of a parabola made?

A

A parabola is the set of all points equidistant from point, called the focus, and a line, called the directrix.

73
Q

thinking of the quadratic formula, why is there only one real root when the discriminant is zero?

A

because the square root of zero is zero, so you are going to +/- 0. The quad formula reduces to -b/2a giving just one root…(the AOS)

74
Q

f(x)= 9x2-6x+3

What is the coefficient of the quadratic term?

A

9

this is also known as

“a”

75
Q

Why is it called a “linear” term

A

alone its graph would be a line, a straight non-horizontal line

75
Q

Projectile motion:

If you are answering a “when” question, what are you solving for?

A

time… to find it you substitute in something for h(t) and solve the quadratic for t

76
Q

If the discriminant is zero, how many times does the parabola hit the x axis?

A

1

78
Q

If “a” is not equal to one when factoring, what should you do?

A

bridge method or

look fordiff of squares

or

perfect square trinomials

79
Q

f(x)= 3x2-6x+3 What is the axis of symmetry?

A

-b/2a = -(-6)/2(3) = 6/6 = 1

so, the axis of symmetry is

x=1

(it is a line)

80
Q

f(x)= -4x2 + 8 What is the axis of symmetry?

A

x= -b/2a = -0/2(-4) = 0/8 = 0. AOS

x=0

82
Q

When completing the square and “a” is not equal to 1, what do you have to do?

A

divide everything by that “a” to turn “a” into “1”

83
Q

f(x)= 6x2- 5x

What is the axis of symmetry?

A

x= -b/2a = -(-5)/2(6) = 5/12,

so, the AOS is

x= 5/12

84
Q

If the discriminant is negative, how many times does the parabola hit the x axis?

A

0

85
Q

f(x)= 6x2- 7x

What is a, b and c ?

A

a = 6, b=7, c = 0

86
Q

where is the vertex located?

A

on the axis of symmetry

(top or bottom)

87
Q

Projectile motion:

how do you find height after 3 seconds?

h(t)= -16t2 + 32t +100

A

plug 3 into function

89
Q

How do you find the vertex of a parabola?

A

the x value is always -b/2a . To get the y value for it, put that x value into the function and get the y. the vertex is at (x,y)

90
Q

f(x)= 3x2-6x+3 What is the quadratic term?

A

3x2

91
Q

How do you graph a parabola?

A

first graph the Axis of symmetry with dotted line, then use that x value to find the vertex and a couple near it. Put those into the function to find points. Reflect over the AOS.

92
Q