Unit 2 - Quadratics Flashcards
factor a² - b² - what are the factors called? what method is this?
(a-b)(a+b) - these are conjugates and the method is difference of squares
what do you write if it has no factors?
Prime. Also please don’t unless you’ve tried absolutely everything. Including imaginary solutions.
What is the zero product property?
If A and B are both real or algebraic expressions, if AB = 0, A or B = 0 (at least one - could be both)
What is the domain of a quadratic function?
set of real numbers, (negative infinity, infinity)
When |a| < 1, does the graph get thiccer or thinner (wider or narrower)
thiccer and wider - a smol = thicc graph
Standard form of a parabola
y = ax² + bx + c
Factored/intercept form of a parabola
y=a(x-p)(x-q)
Vertex form of a parabola
y=a(x-h)²+k
Y intercept of standard form parabola
0, f(0)
x intercepts of a standard form parabola
you gotta factor that spit to get there - no easy peasy formula
Axis of symmetry of a standard form parabola
-b/2a
Vertex of a standard form parabola
-b/2a, f(-b/2a)
Range of a standard form parabola
consider y coord of vertex and sign of a, then make a decision
y intercept of a factored form parabola
0, f(0)
x intercepts of a factored form parabola
(p,0) and (q,0) (please remember to make the sign correct)
AoS of a factored form parabola
(p+q)/2
(find the average/midpoint between x intercepts)
Vertex of a factored form parabola
p+q … p+q
____ , f( _______)
2 … 2
y intercept of a vertex form parabola
0, f(0)
x intercepts of a vertex form parabola
this is very annoying because it’s very sneaky about x intercepts and will not tell you in a simple la dee da formula way
axis of symmetry in a vertex form parabola
h (NOT K FOR THE LOVE OF GOD)
vertex of a vertex form parabola
(h,k) that’s why it’s called vertex
how do you convert vertex form to standard form?
square whatever’s in the parentheses
distribute a
add k
add it all up
how do you convert vertex form into factored form?
by putting it into standard form
and then factoring because math would like to give me a personal f u
If the vertex is (2,4) and a point is (1,3), write a parabola in vertex form that contains those points.
y=a(x-2)²+4
3=a(1-2)²+4
3=a+4
a=-1
y=-(x-2)²+4
What is i?
square root of negative 1
what is the standard form of a complex number?
a+bi
4+5i
4/5 + 3/5 i (don’t use diagonal fractions or Ms. Cooley will breathe fire)
what is sqrt -7 squared?
-7! NOT 7.
x²/8 = -4
x² = -32
x = + or -/-32 (it’s the best thing i have)
x = + or - 4i/2
what is the complex conjugate of a + bi?
a - bi
What is the power order of i?
i if R1, -1 if R2, -i if R3, 1 if no R (r4)
what is the reciprocal of a complex number like -6i?
1/-6i but you must rationalize so multiply by 6i/6i and get 6i/-36i² or 6i/36 or i/6
What is the first step of completing the square?
move c to the other side
ax²+bx+c=0 –> ax²+bx=-c
what is the second step of completing the square?
take half of b and square it and add it to both sides of the equation
ax² + bx + 0.5b² = -c + 0.5b²
what is the 3rd step of completing the square?
factor the polynomial on the right because now it will do so easily and simplify the left
(assume a is a square number - if not just root it)(here i’m using a=1 and b=2 and c=10)
(x+1)² = -9
what is the fourth step of completing the square?
sqrt both sides
/(x+1) = + or - /-9
what is the fifth step of completing the square?
solve for x
x+1 = + or - 3i
x = 1+3i or 1-3i
what is the 6th step of completing the square?
marvel at your genius solutions. idk it’s obsolete ok
what makes a quadratic good for completing the square?
1 (or at least a square number) = a and b = even but it works regardless
what is the discriminant of a quadratic equation?
b²-4ac
if the discriminant is positive, the quadratic has __ real solutions
2
if the discriminant is negative, the quadratic has __ real solutions
0
if the discriminant is 0, the quadratic has __ real solutions
1
sum of quadratic equation roots
-b/a
product of quadratic equation roots
c/a
if given the roots of a quadratic equation, how many possible parabolas could there be?
infinite. just go with a=1 to make your life easier. change it if you need to.
if you complete the square for ax squared + bx + c, what do you get?
quadratic formula (i definitely derived it myself without help from mr lee)
