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Quadratics 5
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Gravity
Terms in this set (96)
ax²+bx+c=0
What is standard form?
(-b±√b²-4ac)/2a
What is the full quadratic formula?
Where the parabola crosses the x-axis
What is the answer to a quadratic equation?
No real solution (imaginary)
What is the answer to a graph if the parabola doesn't cross the x-axis?
The discriminant is positive
How do you know if the answer of a graph has 2 possible answers?
A complex number is a number in the form....
a + bi
(a+bi)(a-bi) =
a^2 + b^2
i^0
1
How to find i to any power, example i^235
divide 235 by 4 and use the remainder, 235 divided by 4 has remainder 3=i^3=-i
What is the complex conjugate of 8-4i
8+4i
how to solve (2i+3)/(9-i)
Multiply the numerator and denominator by the complex conjugate of the denominator which is (9+i)
How to multiply complex numbers for example (2+i)(2i-3)
FOIL Distribute or multiply everything
How to add and subtract complex numbers
combine like terms
i =
The square root of -1
i^2 =
-1
i^3 =
i^2 * i = -1 * i = -i
i^4 =
i^2 * i^2 = -1 * -1 = 1
i^5
i^4 * i = i
i^6
i^5 * i = i^2 = -1
Formula for a complex number:
a + bi
What are the two parts of a complex number?
The real part, and the imaginary part
Discriminant : 4
2 real solutions
x² -4x = -3
Discriminant: 16
2 real solutions
x² + 12 = 8x
2x² + 3x -5 = 0
Discriminant: 49
2 rational solutions
Discriminant: 5
2 real solutions
x² = 3x - 1
Discriminant: -8
No Real Sol...
2 complex solutions
x² -6x+11 = 0
Discriminant: -16
No Real Sol...
2 complex solutions
x² = 2x -5
Vertex
(h,k) Opp. of h
a>0
opens up
a<0
opens down
AOS
x=h
lal>1
narrower
lal<1
wider
+h
left
-h
right
+k
up
-k
down
(-3, -1) down max of -1 Range (-inf, -1]
y=-4(x+3)²-1 What is the vertex, open up or down? Min or max? Range?
ƒ(x)=-2(x+3)²+1 vertex, narrow or wide
(-3, 1) narrow and "upside down" (reflected around x-axis)
(2, -1) up min of -1 Range [-1, inf)
ƒ(x)=x²-4x+3 What is the vertex, open up or down? Min or max? Range?
y=3-4x-x²
Parabola
the graph of a quadratic function
Standard Form
y=ax^2+bx+c
Vertex
the maximum or minimum of the parabola (an ordered pair where the x is always the axis of symmetry and the y is the function evaluated at the a.o.s) - the y is the MIN or MAX
Maximum
the highest point of the parabola (a<0) It opens up.
Minimum
the lowest point of the parabola (a>0). It opens down.
Axis of symmetry
an imaginary line passing through the vertex of the parabola (x=-b/2a)
x-intercepts
point(s) where parabola crosses the x-axis also known as roots, solutions or zeros. Substitute zero in for y and solve for x. It is written as an ordered pair.
y-intercepts
point where parabola crosses the y-axis. "c" of the standard form or plug in x=0 and solve for y.
graphing quadratics
1. find the vertex (you may need to use -b/2a to find x) 2. Make an x/y chart by putting your vertex in the middle and surrounding it
quadratic solutions
at most 2 real solutions depending where the parabola touches the x-axis
x = -5, 5
x² - 25 = 0
x = -7/2, 2
2x² + 3x - 14 = 0
x = 1/3, 7/3
9x² - 24x + 7 = 0
x = -5/3, 5/3
9x² - 25 = 0
x = -10, 11
x² - x - 110 = 0
x² + 7x +12 = 0
x=-3, -4
x² - 81 = 0
x=-9, 9
x² - 7x = 0
x=0, 7
4x² + 7x + 3 = 0
x=-3/4, -1
9x² + 45x = 0
x=0, -5
8x² + 2x - 15 = 0
x=-5/4, 3/2
x² + 10x + 25 = 0
x=-5
4x² - 49 = 0
x=-7/2, 7/2
3x²-5x-1=0
x=5/6 ± √37/6
3x²-9x+1=0
x=(9±√69)/6
3x²-6x=1
x= (3±2√3)/3
2x²+9x-1=0
x=9/4±√(89)/4
x²+8x-13=0
x=-4±√29
(x+6)²-5=0
x=-6±√5
x²-8=0
x=±2√2
(4x+3)²-6=0
x=(-3±√6)/4
(3x-5)²=2
x=(5±√2)/3
4x²-1=8
x=±3/2
2x²+5=37
x=±4
x=2, x=3
Solve x²-5x=-6
x=-2. x=3
Solve x²-x=6
x=-1, x=-5
Solve x²+6x=-5
x=-1, x=5
Solve x²-4x-5=0
x=-1, x=7
Solve x²-6x-7=0
x=-7, x=-1
Solve x²+8x+7=0
x=5
Solve x²-10x=-25
x=4
Solve x²-8x+16=0
x=-3
Solve x²+6x=-9
x=3
Solve x²=6x-9
x=-3, x=3
Solve x²=9
x=-2, x=2
Solve x²-4=0
x=±√5
Solve x²-5=0
x=±√6
Solve x²=6
y=a(x-h)^2+k
What is vertex form of a quadratic?
(3, -4)
What is the vertex of y=2(x-3)^2-4
y=a(x-h)^2+k
What is vertex form of a quadratic?
(3, -4)
What is the vertex of y=2(x-3)^2-4
y= 2x^2-4x+1
What is the quadratic model that includes the points (0,1) (2,1) (4,17)?
x=(2±i√41)/3
What are the solutions to 0=3x^2-4x+15
x=(5±i√47)/6
Solve: 3x^2=5x-6
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