Projectile motion:

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

h(t)= -16t^{2} + 44t + 95

the time at the peak is

t=-b/2a

plug in b and a to find t

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

just factor out the x

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

What is the zero product property?

(this)(that)=0

is true when

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

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

f(x)= 3x^{2}-4x+5 What is a, b and c ?

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

How do you solve by isolating the x^{2} ?

When there is no linear term (no x term), simply act like you’re solving for x, but solve for x^{2} and in the last step, take square root of both sides ( + / - )

f(x) = ax^{2} + bx + c

What is the quadratic term?

ax^{2}

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

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

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

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

what completes the square and what is the factored form? x^{2} + 5x + ____

25/4 completes it

and

(x + 5/2) ^{2}

^{is factored form}

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

2

Projectile motion:

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

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

Describe what an axis of symmetry is?

it is where the parabola can fold onto itself

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

NO

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

crosses x axis at those roots, at those x values

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

YES

why is it called a constant term

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

What are solutions to

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

x = { -8, 3}

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

What do we call b^{2}-4ac

discriminant

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

2

When can you just isolate the x^{2} and root both sides?

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

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

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.

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

there is

one real double root

f(x)= -4x^{2} + 8

What is a, b and c ?

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

What does the discriminant tell us about?

about the roots

whether they are real or not

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

h(t)= -16t^{2} + 48t +350

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

Why is it called a “quadratic?”

From the latin “quadrus” which means “square”

h(t)= -16t^{2} + 55t - 60

what is the “-16”

that is the pull of gravity, actually, half of gravity. Gravity pulls at 32ft/sec^{2} at sea level.

what completes the square and what is the factored form? x^{2} + 6x + ____

9 completes it and (x+3)^{2}

How do you solve by completing the square?

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

f(x)= 3x^{2}-6x+5

What is the coefficient of the constant term?

5

this is also known as

“c”

What are the five ways to solve a quadratic?

isolate x^{2}, factor, complete square, quadratic formula, graph and look for x-intercepts

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

the roots, the zeros, the solutions

how would you solve

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

(what are solutions?)

set factors = 0 and solve:

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

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

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

0

Projectile motion:

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

h(t).

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

h(t)= -16t^{2} + 77t +45 what is the initial velocity?

77 ft/sec

f(x)= 2x^{2}-6x+3

What is the coefficient of the linear term?

-6

this is also known as

“b”

f(x) = ax^{2 }+ bx + c

What is the linear term?

bx

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

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

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

h(t)= -16t^{2} + 32t +100

plug 10 into function

(where the t’s are)

and you’ll find h(t)

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

h(t)= -16t^{2} + 32t +100

Either trying to find

height, h(t),

or time, t

h(t)= -16t^{2} + 25t +100 what is the “25”

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

Projectile motion:

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

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

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

doesn’t touch x axis

h(t)= -16t^{2} + 12t +10 what is the “10”

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

f(x)= 3x^{2}-6x+3 What is the constant term?

3

it is the y intercept

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

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

Projectile motion:

how do you find the maximum height?

h(t)= -16t^{2} + 32t +100

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

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

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

What is the discriminant?

b^{2}-4ac

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

Projectile motion:

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

h(t)= -16t^{2} + 32t +100

put 40 where h(t) is.

Solve quadratic for t

h(t)= -16t^{2} + 32t +600

what is the y intercept?

600, It is the initial height

why is it called a discriminant?

because it discriminates between which type of roots you will have

Projectile motion:

how do you find the time it hits the ground?

h(t)= -16t^{2} + 88t +100

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

h(t)= -16t^{2} + 66t +100 what is the initial height?

100 ft

How do you know when it is a quadratic?

when there is an x^{2} term (quadrus:square)

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

there are no real roots,

2 imaginary roots,

it doesn’t touch x axis

f(x)= 3x^{2}-6x+3 What is the linear term?

-6x

f(x) = ax^{2} + bx + c

What is the constant term?

c

(it is the y intercept)

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

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

what completes the square and what is the factored form? x^{2} - 8x + ____

16 completes it and (x-4)^{2}

How do you solve by factoring?

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

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

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

why are roots called “zeros”

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

f(x)= 3x - 9

What is a, b and c?

a = 0 , b = 3, c = 9

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

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

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

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

what completes the square and what is its factored form? x^{2} + b/a x + ____

b^{2}/4a^{2} completes it

and

(x+ b/2a)^{2}

what is a coefficient?

the number in front of a variable or variables

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

1

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

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

Geometrically, how is the shape of a parabola made?

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

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

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)

f(x)= 9x^{2}-6x+3

What is the coefficient of the quadratic term?

9

this is also known as

“a”

Projectile motion:

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

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

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

1

Why is it called a “linear” term

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

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

bridge method or

look fordiff of squares

or

perfect square trinomials

f(x)= 3x^{2}-6x+3 What is the axis of symmetry?

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

so, the axis of symmetry is

x=1

(it is a line)

f(x)= -4x^{2} + 8 What is the axis of symmetry?

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

x=0

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

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

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

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

f(x)= 6x^{2}- 5x

What is the axis of symmetry?

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

so, the AOS is

x= 5/12

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

0

f(x)= 6x^{2}- 7x

What is a, b and c ?

a = 6, b=7, c = 0

where is the vertex located?

on the axis of symmetry

(top or bottom)

Projectile motion:

how do you find height after 3 seconds?

h(t)= -16t^{2} + 32t +100

plug 3 into function

(instead of t), fint h(t)

What is the equation of the axis of symmetry?

x = -b / 2a

How do you find the vertex of a parabola?

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)

f(x)= 3x^{2}-6x+3 What is the quadratic term?

3x^{2}

How do you graph a parabola?

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.

What is the quadratic formula?

x = -b +- sqroot b squared

- 4ac all over 2a

(but in formula)