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| Numbers that can be expressed as a fraction |
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| ((b^2)-(4ac)) ^ 1/2 (aka the square root) |
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i = square root of -1
i^2 = -1
i^3 = -i
i^4 = 1 |
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| Point Slope Form aka Slope Intercept Form |
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{(x2+x1)/(2)} , {(y1+y2)/(2)}
[This equals {x},{y} or the midpoint, fyi.] |
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((-b)+-((b^2)-4ac)^1/2)/2*a
[This uses the square root.] |
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1. Write coefficients in order
2. Start with leading coefficient
3. Multiply by the value of x
4. Add the next coefficient |
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| Review: Factors & Multiples (from Factors & Multiples WS #1) |
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| 5 is a factor of 10, 10 is a multiple of 5, 1 is not prime, 2 is the only even prime #. |
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| Remainder Theorem (in Synthetic Substitution) |
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| When a polynomial is divided by x-a, the remainder = p(a) |
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| Factor Theorem (in Synthetic Substitution) |
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| x-a is a factor IF p(a)=0 |
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| Graphing Polynomial Functions: if a>0, aka if a=positive.... (First degree is 3.) |
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The left and right ends of the graph go upwards.
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/\ /
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/ |
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| Graphing Polynomial Functions: if a<0, aka if a=negative.... (First degree is 3.) |
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The left and right ends of the graph go downwards.
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\ /\
\/ \
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| Graphing Polynomial Functions: if a>0, aka if a=positive.... (First degree is 4.) |
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Both the left and right ends go upward:
\ /
\ /\ /
\/ \/ |
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| Graphing Polynomial Functions: if a<0, aka if a=negative.... (First degree is 4.) |
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Definition
Both the left and right ends go downward:
/\ /\
/ \/ \
/ \ |
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| Double Root (ie x^2 = 0 & 0 roots) |
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| This means that the graph is tangent to the x-axis; the graph does not touch the x-axis |
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| (See 2.6 notes, this gets confusing on computer) Solving Polynomials w/ Factoring: p's & q's: p(x)=(a_n_x^n)+ (a_n-1_x^(n+1))+....+a_0_ |
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x={p/q}, meaning:
p are the factors of a_0
q are the factors of a_n |
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| {Sum of terms}/{Number of Terms} |
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| The number that appears most often |
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1. Find the zeroes
2. Perform sign analysis (to figure out if it's positive or negative)
3. Determine the Answer |
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| Absolute Value Inequalities |
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1. Remove absolute value signs and solve (positive part)
2. Switch the inequality around and change the sign of every other part on the other side (negative part)
3. Check for extraneous solutions |
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| Inequalities in 2 Variables |
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1. Graph both inequalities
2. For each equation, pick a point to determine where to shade.
3. The overlap of shading = the answer |
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A function is a rule that assigns to every element in a set D exactly one element set R.
D is the domain (x)
R is the range (y)
Zeroes: Where the graph crosses the x-axis |
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| 1. If no vertical line intersects a graph in more than one point, then the graph represents a function |
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(fog)(x) = f(g(x))
x is the domain of g, and g(x) is the domain of f. |
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| Reflect across the x-axis or y-axis. |
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| Reflecting Graphs: Absolute Value |
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| A reflecting graph, only in the positive areas. |
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Switch y and x.
ie: y=x^2 would be x=y^2 |
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Fundamental period: The smallest period of a periodic function
Amplitude: Half of the difference between a periodic function's maximum and minimum points (equation = (max-min)/(2) |
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y-k=f(x-h)
k = number on x-axis, move to right if k is positive
h = number to move on the y axis, move upward is h is positive |
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f(.5x) = stretch
f(2x) = shrink
cf(x) = vertical
f(cx) = horizontal |
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| Law of Exponents: Same Bases |
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(b^x)(b^y) = b^(x+y)
(b^x)/(b^y) = b^(x-y) |
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| Law of Exponents: Same Exponents |
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((ab)^x) = a^x(b^x)
((a/b)^x) = (a^x)/(b^x) |
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| Law of Exponents: Power of a power |
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Diameter: 2r
Area = (PIE)r^2
Circumference = 2(PIE)r |
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Diameter: 2r
Area = (PIE)r^2
Circumference = 2(PIE)r |
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| b^(p/q) = "q" root of b to the "p" power. |
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A(t)=A_0_(1+r)^t
A(t)= A_0_b^(t/k) |
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| If a quantity is growing at r % per year, then the doubling time is 72/r years. |
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ie. log_8_x=a is also 8^a=x
natural log = lne |
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Add: log_b_MN=log_b_M+log_b_N
Subtract: log_b_(M/N)=log_b_M-log_b_N
log_b_M^N = Nlog_b_M |
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A function that contains a variable in exponent
ie 2^(x-3)=8 |
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| log_b_C = (log_a_C)/(log_a_B) |
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((x-h)^2)+((y+k)^2)=r^2
Point: x,y
Center: h,k |
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| Intersection of Line/Circle |
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1. Solve linear system in terms of y
2. Substitute this value into the circle equation. Solve.
3. Take the answers from step 2 and plug into the linear equation. |
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| Ellipses (not going to focus much on these, since it was the last topic covered) |
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{(x^2)/(a^2)} + {(y^2)/(b^2)} = 1
A is always the bigger number
If A is underneath the x, it's a horizontal graph. If A is underneath the y, it's a vertical graph. |
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