BASIC QUADRATIC FACTS Flashcards
What is the equation of the axis of symmetry?
x = -b / 2a
What are the roots of a function? (other names?)
the zeros, the x-intercepts, the solutions, where the parabola crosses
Projectile motion:
How do you find at what time it is at its peak?
h(t)= -16t2 + 44t + 95
the time at the peak is
t=-b/2a
plug in b and a to find t
f(x)= 3x - 9
What is a, b and c?
a = 0 , b = 3, c = 9
What are the zeros of a function? (other names?)
the roots, the x intercepts, the solutions, where the parabola crosses
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.
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)= 3x2-4x+5 What is a, b and c ?
a = 3, b = -4, c = 5
what completes the square and what is the factored form? x2 + 5x + ____
25/4 completes it
and
(x + 5/2) 2
is factored form
How do you solve by isolating the x2 ?
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 ( + / - )
f(x) = ax2 + bx + c
What is the quadratic term?
ax2
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
f(x)= 3x2-6x+3 What is the constant term?
3
it is the y intercept
f(x)= 2x2-6x+3
What is the coefficient of the linear term?
-6
this is also known as
“b”
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
why is it called a constant term
alone it would be a constant function with a constant y-value (height)
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
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 b2-4ac
discriminant
what completes the square and what is the factored form? x2 + 6x + ____
9 completes it and (x+3)2
If the discriminant is positive, how many times does the parabola hit the x axis?
2
When can you just isolate the x2 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.
f(x)= -4x2 + 8
What is a, b and c ?
a = -4, b = 0, c = 8
If the discriminant is zero, what do you know about roots?
there is
one real double root
Why is it called a “quadratic?”
From the latin “quadrus” which means “square”
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)= -16t2 + 48t +350
put 10 where h(t) is. Solve quadratic for t
h(t)= -16t2 + 55t - 60
what is the “-16”
that is the pull of gravity, actually, half of gravity. Gravity pulls at 32ft/sec2 at sea level.
What are the five ways to solve a quadratic?
isolate x2, factor, complete square, quadratic formula, graph and look for x-intercepts
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)= 3x2-6x+5
What is the coefficient of the constant term?
5
this is also known as
“c”
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)= -16t2 + 77t +45 what is the initial velocity?
77 ft/sec
f(x)= 5 - 2x +3x2 What is a, b and c ?
a = 3, b = -2, c = 5
f(x) = ax2 + bx + c
What is the linear term?
bx
Projectile motion: How do you find the height after 10 seconds?
h(t)= -16t2 + 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)= -16t2 + 32t +100
Either trying to find
height, h(t),
or time, t
h(t)= -16t2 + 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)= -16t2 + 12t +10 what is the “10”
that is the initial height, or starting height, y intercept. 10 ft.
What is the discriminant?
b2-4ac
(notice there is no sq root on it!!)
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)= -16t2 + 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
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
Projectile motion:
How do you find the time when it is 40 ft high?
h(t)= -16t2 + 32t +100
put 40 where h(t) is.
Solve quadratic for t
h(t)= -16t2 + 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)= -16t2 + 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)= -16t2 + 66t +100 what is the initial height?
100 ft
How do you know when it is a quadratic?
when there is an x2 term (quadrus:square)
f(x)= 3x2-6x+3 What is the linear term?
-6x
f(x) = ax2 + bx + c
What is the constant term?
c
(it is the y intercept)
what completes the square and what is the factored form? x2 - 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.
why are roots called “zeros”
because it is where the y value is zero, (the height of function is zero)
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? x2 + b/a x + ____
b2/4a2 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)= 9x2-6x+3
What is the coefficient of the quadratic term?
9
this is also known as
“a”
Why is it called a “linear” term
alone its graph would be a line, a straight non-horizontal line
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
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)= 3x2-6x+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)= -4x2 + 8 What is the axis of symmetry?
x= -b/2a = -0/2(-4) = 0/8 = 0. AOS
x=0
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)= 6x2- 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)= 6x2- 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)= -16t2 + 32t +100
plug 3 into function
(instead of t), fint h(t)
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)= 3x2-6x+3 What is the quadratic term?
3x2
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)
