Sabrehalrubeai,
In order to find the correct answer to this question we need to go over all of the options to see which one is the correct answer.
Therefore your answer is option A "ratio."
Hope this helps!
<em>*To solve for a variable, isolate the desired variable onto one side.</em>
Firstly, we want to add 1/3 to each side however -5/6 and 1/3 do not share the same denominator, and we want them to have that and we can do that by finding the LCD, or lowest common denominator. To find the LCD, list the multiples of 6 and 3 and the lowest multiple that they share is their LCD. In this case, their LCD is 6. Multiply -1/3 by 2/2:
Now that we have common denominators, add both sides by 2/6:
Next, you want to cancel out 2 to isolate z. Usually, one would divide both sides by 2, however remember that <u>dividing by a number is the same as multiplying by it's reciprocal.</u> To find a number's reciprocal, flip the numerator and denominator around. In this case, since 2 is a <em>whole number</em> this means that the denominator is 1. In this case: 2/1 would become 1/2. Multiply both sides by 1/2:
<u>Your final answer is z = -1/4.</u>
NOTES:
Degree: the largest exponent in the polynomial
End Behavior:
Multiplicity (M): the exponent of the zero. <em>e.g. (x - 3)² has a multiplicity of 2</em>
Relative max/min: the y-value of the vertices.
Rate of Change: slope between the two given points.
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1. f(x) = (x-1)²(x + 6)
a) Degree = 3
b) end behavior:
c) (x - 1)²(x + 6) = 0
x - 1 = 0 x + 6 = 0
x = 1 (M=2) x = -6 (M=1)
d) The midpoint between 1 and -6 is -3.5, so axis of symmetry is at x = -3.5
y = (-3.5 - 1)²(-3.5 + 6)
= (-4.5)²(2.5)
= 50.625
50.625 is the relative max
e) see attachment #1
f) The interval at which the graph increases is: (-∞, -3.5)U(1, ∞)
g) The interval at which the graph decreases is: (-3.5, 1)
h) f(-1) = (-1 - 1)²(-1 + 6)
= (-2)²(5)
= 20
f(0) = (0 - 1)²(0 + 6)
= (-1)²(6)
= 6
Find the slope between (-1, 20) and (0, 6)
m =
=
= -14
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2. y = x³+3x²-10x
= x(x² + 3x - 10)
= x(x + 5)(x - 2)
a) Degree = 3
b) end behavior:
Coefficient is positive so right side goes to positive infinity
Degree is odd so left side goes to negative infinity
c) x(x + 5)(x - 2) = 0
x = 0 x + 5 = 0 x - 2 = 0
x = 0 (M=1) x = -5 (M=1) x = 2 (M=1)
d) The midpoint between -5 and 0 is -2.5, so axis of symmetry is at x = -2.5
y = -2.5(-2.5 + 5)(-2.5 - 2)
= -2.5(2.5)(-4.5)
= 28.125
28.125 is the relative max
The midpoint between 0 and 2 is 1, so axis of symmetry is at x = 1
y = 1(1 + 5)(1 - 2)
= 1(6)(-1)
= -6
-6 is the relative min
e) see attachment #2
f) The interval at which the graph increases is: (-∞, -2.5)U(1, ∞)
g) The interval at which the graph decreases is: (-2.5, 1)
h) f(-1) = -1(-1 + 5)(-1 - 2)
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3. y = -x(x + 2)(x - 7)(x - 3)
a) Degree = 4
b) end behavior:
Coefficient is negative so right side goes to negative infinity
Degree is even so left side goes to negative infinity
c) -x(x + 2)(x - 7)(x - 3) = 0
-x = 0 x + 2 = 0 x - 7 = 0 x - 3 = 0
x = 0 (M=1) x = -2 (M=1) x = 7 (M=1) x = 3 (M=1)
d) The midpoint between -2 and 0 is -1, so axis of symmetry is at x = -1
y = -(-1)(-1 + 2)(-1 - 7)(-1 - 3)
= 1(1)(-8)(-4)
= 32
32 is a relative max
The midpoint between 0 and 3 is 1.5, so axis of symmetry is at x = 1.5
y = -(1.5)(1.5 + 2)(1.5 - 7)(1.5 - 3)
= -1.5(3.5)(-5.5)(-1.5)
= -43.3125
-43.3125 is the relative min
The midpoint between 3 and 7 is 5, so axis of symmetry is at x = 5
y = -(5)(5 + 2)(5 - 7)(5 - 3)
= -5(7)(-2)(2)
= 140
140 is the relative max
e) see attachment #3
f) The interval at which the graph increases is: (-∞, -1)U(1.5, 5)
g) The interval at which the graph decreases is: (-1, 1.5)U(5, ∞)
h) f(-1) = -(-1)(-1 + 2)(-1 - 7)(-1 - 3)
= 1(1)(-8)(-4)
= 32
f(0) = -(0)(0 + 2)(0 - 7)(0 - 3)
= 0
Find the slope between (-1, 32) and (0, 0)
m =
=
= -32
