Algebra 2 Concepts Flashcards
Why are they called “quadratics?”
It comes from the Latin word “quadrus” which means SQUARE
What is the tough thing about quadratics?
The fact that they are squared.. We have to undo that x2. We want to get it linear!!
How do we handle that squared term? how can we undo it?
By either splitting the x’s apart by factoring, or by taking a square root somewhere..
Why is it called a “linear” term
It creates a straight line when graphed
Why is it called a constant term
Alone it would be a constant function.. The same height always. a horizontal line with a constant y, a constant height
Why is it called a discriminant?
Because it discriminates between which type of solutions, roots and zeros you will have (2real, one real or imaginary)
Are all quadratics similar?
YES.. THEY ALL HAVE THE EXACT SAME SHAPE.. They are simply zoom-ins or zoom-outs of eachother..
Why do parabolas look different if they are all similar?
Well…. Simply either imagine zooming way in or zooming way out… this method can always get any 2 parabolas to look the same.
f(x)= 3x2-4x+5 What is a, b and c ?
a = 3, b = -4, c = 5
How do you know when it is a quadratic?
When the highest degreed term is the x2 term (quadrus:square)
f(x)= 3x2-6x+3 What is the quadratic term?
3x2
f(x)= 3x2-6x+3 What is the linear term?
-6x
f(x)= 3x2-6x+3 What is the constant term?
3, it is the also the y intercept
What is a coefficient?
The number in front of a variable or variables
f(x)= 3x2-6x+3 What is the coefficient of the quadratic term?
3 this is also known as “a”
f(x)= 3x2-6x+3 What is the coefficient of the linear term?
-6 this is also known as “b”
f(x)= 3x2-6x+3 What is the coefficient of the constant term?
3 this is also known as “c”
What is the quadratic term’s exponent on the x ?
2
What is the linear term’s exponent on the x ?
1
What is the constant term’s exponent on the x?
0
f(x) = ax2 + bx + c What is the coefficient of the quadratic term?
a
f(x) = ax2 + bx + c What is the coefficient of the linear term?
b
f(x) = ax2 + bx + c What is the coefficient of the constant term?
c (it is the y intercept)
f(x) = ax2 + bx + c What is the quadratic term?
ax2
f(x) = ax2 + bx + c What is the linear term?
bx
f(x) = ax2 + bx + c What is the constant term?
c (it is the y intercept)
Describe what an axis of symmetry is
it is where the parabola can fold onto itself. It is the equation of this vertical line, so you write “x = ___”(-b/2a)
What is the equation of the axis of symmetry?
x = -b / 2a(notice it is the equation of a vertical line) (also Notice it is hidden in the quadratic formula)
Where is the vertex located?
On the axis of symmetry (highest or lowest point)
What is the quadratic formula?
x = -b +- sqroot b squared - 4ac all over 2a (but in formula)
What is the discriminant?
b2-4ac (notice there is no sq root on it!!)
What does the discriminant tell us about?
About the roots, the zeros, how many x intercepts the parabola has, how many times it crosses the x axis, whether roots are real or imaginary
What do we call b2-4ac
The discriminant
Why are roots called “zeros”
Because it is where the height is zero; where the y value is zero; they are the x values that make the y=0
What is difference between (3, -2) and {3, -2}
The first is a point on a plane (over 3, down 2) and is in parentheses… The second set are numbers in braces, so it is simply a set containing two values, 3 and -2. It is not a point on a plane, the order does non matter in braces.
What are the roots of a function? (other names?)
The zeros, the x-intercepts, the solutions,where the parabola crosses
What are the zeros of a function? (other names?)
The roots, the x intercepts, the solutions, where the parabola crosses
What are the solutions of a function? (other names)
The roots, the x intercepts, the zeros where the parabola crosses
What are the x intercepts of a function? (other names?)
The roots, the zeros, the solutions
What are the three common forms of quadratic functions?
Standard form, vertex form, and factored form (factored form can be called zero form or root form)
What is the Vertex form of a quadratic?
f(x) = a(x-h)2 + k
What is the Factored form of a quadratic?
f(x)=a(x-r)(x-r)
what is tricky about the Vertex form?
the -h part.. If its (x+9)2 +8 then h is -9.. remember that h is also the AOS !!!
What is the Standard form of a quadratic?
ax2+bx+c = 0
f(x) = a(x-h)2 + k How do we know if it opens up or down?
by the sign of a, if its positive, up, negative , down
How do you know when a parabola is smiling?
when a is positive
How do you know when a parabola is frowning?
when a is negative
How is the axis of symmetry hidden in the quadratic formula?
it is the beginning.. (put both parts over 2a)
Do all quadratic functions have x intercepts?
No, some don’t touch the x axis.. These have imaginary roots and a negative discriminant
Do all quadratics have a y intercept
YES… All quadratics have exactly 1 y-intercept.. (c).. if there is no constant term, the y intercept is 0, so the parabola passes through the origin!!!!
What are the three forms of a quadratic function?
Standard y=ax2+bx+cVertex y=a(x-h)2+kFactored y=e(x+f)(x+g)
If (x-7) is a factor then _____ is a zero (or root)
7
if -8 is a zero then ___ is a factor
(x+8)
If 9 is a zero then___ is a factor
(x-9)
If (x+6) is a factor then ____ is a zero (or root)
-6
If (x-8) is a factor then ____ is an x intercept.
(8,0)
If 12 is a zero, then ___ is an x intercept
(12,0)
If (4,0) is an x intercept, then ____ is a factor
(x-4)
if (-9, 0) is an x intercept, then ___ is a zero (or root)
-9
Be careful: if (2x - 1) is a factor, then ___ is a zero
1/2 You have to solve 2x-1= 0
Be careful: if 2/3 is a root then ______ is a factor
(x-2/3) OR (3x-2)
What does “a” tell us in all forms of quadratics?
“+a” smiling (min), and “-a” frowning (max)
What graphing gift is given when you get a quadratic in standard form? y=ax2+bx+c?
The y intercept: (0,C). To get aos use x=-b/2a
What graphing gift is given when you get a quadratic funciton in factored form? y=(x-r1)(x-r2)
The ZEROS, the x-intercepts are the roots. (r1, 0) and (r2, 0) are the x intercepts. The AOS is between them!
What graphing gift is given when you get a quadratic funciton in factored form? y=a(x-h)2+k
TWO GIFTS! THE VERTEX and the AOS! vertex at (h,k). Don’t forget opposite of just h. The AOS is x=h
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.
f(x)= 3x2-6x+3 What is the axis of symmetry?
-b/2a = -(-6)/2(3) = 6/6 = 1
f(x)= 5 - 2x +3x2 What is a, b and c ?
a = 3, b = -2, c = 5
f(x)= -4x2 + 8 What is a, b and c ?
a = -4, b = 0, c = 8
f(x)= 3x - 9 What is a, b and c?
a = 0 , b = 3, c = 9..There is no quadratic term.. THIS IS LINEAR!!! IT’S A LINE!slope of
f(x)= 6x2- 7x What is a, b and c ?
a = 6, b=-7, c = 0
f(x)= 6x2- 5x What is the axis of symmetry?
x= -b/2a = -(-5)/2(6) = 5/12, AOS x= 5/12
f(x)= -4x2 + 8 What is the axis of symmetry?
x= -b/2a = -0/2(-4) = 0/8 = 0. AOS is x=0, which is the y axis!!
What are the five ways to solve a quadratic?
Set to zero and factor, complete square, quadratic formula, graph and look for x-intercepts, or when there is no linear term, isolate the x2 and root both sides.
When can you just isolate the x2 and root both sides?
When there are no linear terms (only quad and const)
How do you solve by completing the square?
Get all quadratic and linear term on one side and bring constant term over to other…. Make sure a=1, take half of b and square it, add it to both sides. Factor, root, solve
What completes the square and what is the factored form? x2 + 6x + ____
9 completes it and the factored form is (x+3)2
What completes the square and what is the factored form? x2 - 8x + ____
16 completes it and the factored form is (x-4)2
What completes the square and what is the factored form? x2 + 5x + ____
25/4 completes it and the factored form is (x + 5/2) 2
What completes the square and what is its factored form? x2 + b/a x + ____
b2/4a2 completes it and the factored form is (x+ b/2a)2
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 that make the equation true. They are also the x intercepts of the parabola
How do you solve by factoring?
Get stuff all on one side, factor, set each factor = 0, solve each linear equation.
How do you solve by isolating the x2 ? (when can you do it?)
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 ( + / - )
If the discriminant is zero, what do you know about roots?
There is one double root, it is a perfect square, the parabola just touches down x axis without crossing, the vertex is on the x axis.
If the discriminant is negative, what do you know about roots?
There are no real roots, 2 imaginary roots, the parabola doesn’t touch x axis
If the discriminant is positive, what do you know about roots?
There are 2 real roots, and the parabola crosses x axis twice..at these roots
Explain what a parabola with no real roots (real zeros) looks like.
Doesn’t touch x axis
Explain what a parabola with one real root (real zero) looks like.
Just touches x axis at one spot, the root (the zero), it doesn’t cross it, it bounces off it, called a “double root”, the vertex is on the x axis
Explain what a parabola with two real roots (real zeros) looks like.
Crosses x axis at those roots, at those x values
If the discriminant is negative, how many times does the parabola hit the x axis?
0
If the discriminant is zero, how many times does the parabola hit the x axis?
1
If the discriminant is positive, how many times does the parabola hit the x axis?
2
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 number.. .hence.. Imaginary roots (not real)
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 + and - zero… the quad formula reduces to -b/2a .. The only root is the axis of symmetry (the vertex is on the x axis)
Thinking of the quadratic formula, why are there 2 real roots when the discriminant is positive?
Because there is a square root of a positive, you will have to + and - that in the top, giving 2 different real solutions.
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
How do you graph a parabola?
First graph the Axis of symmetry with dotted line, (x= -b/2a) then make a T table and use that x value to find the vertex and use a couple integers near it to find other points. LEAVE SPACE FOR ARITHMETIC. Reflect over the AOS and connect the dots
How do you convert from standard to vertex form?
Leave space, complete square by ADDING ZERO ( +4 -4), write the square as factored form
How do you convert from vertex to standard form?
Square the binomial and combine stuff
f(x) = a(x-h)2 + k where it the vertex? what is the Axis of Symmetry?
Vertex at the point (h, k), the AOS is the vertical line x= h
f(x) = a(x-h)2 + k How do we know if it has max or min?
By the sign of a, if its positive it opens up so it has a MIN, negative it opens down so it has a MAX
f(x)= -2(x-5)2-3What is vertex and is it a min or a max? What is the axis of symmetry?
at ( 5, -3) and it is a maximum, the AOS is x=5
f(x)= -4(x-8)2+6What is vertex and which way does it open?what is AOS?
at (8, 6) and opens downward (has max). The AOS is x=8
f(x)= 2(x+4)2-3 What is vertex and which way does it open?
at (-4, -3) and it opens upwards (has min)
f(x)= (x+9)2-8 What is vertex and is it a min or a max?
at (-9, -8) and it is a minimum
When converting from standard to vertex form, and “a” is not 1, what do you have to be careful about?
Focus on quadratic and linear term.. Factor out “a”.. Complete the square, be sure to subtract (a times the number you used to complete square) from the constant term so that you are adding ZERO
How do you find the vertex of a parabola in standard form?
The x value is always -b/2a . To get the y value for it, put that x value into the function and solve
How do you know if a parabola has a max or minimum by looking at the function in standard form?
if “ a” is pos, it has a min. if ‘a’ is negative, it has a max
How do you know if a parabola in standard form opens upward (smile) or down (frown)?
if “a” is pos it opens up. if a is negative, it opens down. Positive people smile, negative people frown.
How do you find the max or min value of a quadratic in either form?
It is the y value at the vertex. In vertex form, it’s simply k. In standard form, Plug -b/2a into the function to get y.remember to look at “a” + smiles have min.. - frowns have max.
When completing the square and “a” is not equal to 1, what do you have to do?
Divide both sides of equation (every term) by that “a” to turn “a” into “1”
Does “a” have to be equal to 1 when doing quadratic formula?
NO
Does “a” have to be equal to 1 when completing a square?
YES
If there is no linear term, what is an easy way to solve?
Simply isolate the x2 and +/- root both sides.
If “a” is not equal to one and you want to factor, what should you do?
Bridge (ac) method (or look for perfect square trinomial or diff of squares)
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 what we use when solving by factoring.
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. The solutions are x = { 5, 2/3}
how would you solve 4x2 - 9 = 13 ? (just say what you’d do)
No linear term, so just isolate the x2 and root sides
How would you solve 9x2 - 3x = 0 ? (just say what you’d do)
Factor out x and set factors=0.. (one solution will be 0)
What does the quadratic formula say about finding the x intercepts?
Stand at the AOS and reach up and down a bit to find the solutions
What is solution to:5(x-8)(x+3)=0
8 and -3(this)(that)=0 situationthis=0 and that=0we are looking for the x values that make the equation true.. so we want to make x-8=0 and x+3=0
What is solution to:g (x+f)(x-w) = 0
-f and w(this)(that)=0 situationthis=0 and that=0we want to make equation trueso we solve x+f=0 and x-w=0
What are the x intercepts of:y=5(x+6)(x-3)
at -6 and at 3(they are also called solutions or zeros)
What are the zeros of y=5(x-3)(2x+1)
The zeros, or roots, or x intercepts are at(3, 0) and (-1/2, 0)
