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79 Cards in this Set
- Front
- Back
Distance Formula |
▫︎▫︎▫︎▫︎▫︎▫︎_________________ d = √ (x₂- x₁)² + (y₂- y₁)² |
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Midpoint Formula |
⎧ x₁+ x₂ ▫︎▫︎ y₁+ y₂ ⎫ ⎪--------- , ---------⎜ ⎩ ▫︎▫︎▫︎2▫︎▫︎▫︎▫︎▫︎▫︎ 2▫︎▫︎▫︎▫︎▫︎⎭
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How to solve h(x) = g(x) by the Intersection Method |
1. Graph y₁ = h(x) and y₂ = g(x). 2. Find each point of intersection. 3. The solutions are the x-values of each point of intersection. |
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How to solve h(x) = g(x) by the Zero/X-Intercept Method |
1. Rewrite the equation as h(x) - g(x) = 0. 2. Graph y₁ = h(x) - g(x). 3. Find the zeros/x-intercepts. 4. The solutions are the x-values of each zero.
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Standard form of a line |
𝐴𝑥 + 𝐵𝑦 = 𝐶 |
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Slope of a line in standard form |
m = −𝐴 ⁄𝐵
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Slope-Intercept form of a line |
y = mx + b |
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Point-Slope form of a line |
y - y₁ = m( x - x₁) |
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Relation |
A set of ordered pairs |
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Domain |
Set of all input/x-values of a relation |
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Range |
Set of all output/y-values of a relation |
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Function |
A special relation such that each input corresponds to exactly one output. |
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Independent variable |
x |
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Dependent variable |
y |
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If a function contains an even radical, then its domain is evaluated by |
setting the "inside" of the radical ≧ 0 . |
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If a function has a fraction in it (or is a fraction itself), then its domain is evaluated by |
setting the denominator ≠ 0 . |
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Difference Quotient |
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A function is even if for every value of x, |
f ( -x ) = f ( x ) |
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An even function is symmetric with respect to the |
y - axis |
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A function is odd if for every value of x, |
f ( -x ) = - f ( x ) |
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Informally, a function is continuous if |
its graph can be drawn without lifting your pencil from the paper. |
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Vertical shift (with parent function f(x) ) |
g( x ) = f( x ) ± c |
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Horizontal shift (with parent function f(x) ) |
g( x ) = f( x ± c) |
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Reflection over x-axis (with parent function f(x) ) |
g( x ) = - f( x ) |
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Reflection over y-axis (with parent function f(x) ) |
g( x ) = f( -x ) |
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Vertical stretch/compression (with parent function f(x) ) |
g( x ) = c × f( x)
c > 1 stretch 0 < c < 1 compression |
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Horizontal stretch/compression (with parent function f(x)) |
g( x ) = f( c × x )
0 < c < 1 stretch c > 1 compression |
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Two things you must do when solving a radical equation are |
1. isolate the radical 2. check your answers |
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Discriminant of a quadratic |
b² - 4ac
> 0 : 2 real solutions = 0 : 1 real solution < 0 : no real solutions |
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The x-value (h) of the vertex (h,k) of a quadratic (parabola) in polynomial form (ax² + bx + c) can be found by |
▫︎▫︎▫︎▫︎▫︎ - b h = ------ ▫︎▫︎▫︎▫︎▫︎ 2a |
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Definition of absolute value |
▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎⎧x if x ≧ 0 |x| = ▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎ ⎩-x if x < 0 |
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Properties of polynomial functions |
1. Domain: ( -∞, ∞ ) or ℝ 2. Continuous 3. Graphs are smooth & rounded |
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End behavior of an odd function |
Ends go in opposite directions
a > 0 down on right, up on left a < 0 up on right, down on left |
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End behavior of an even function |
Ends go in same direction
a > 0 up a < 0 down |
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A polynomial function of degree n has at most ___ zeros. |
n |
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A polynomial function of degree n has at most ___ turning points/local extrema/local max/min. |
n - 1 |
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A polynomial function of degree n has at most ___ points of inflection (a point of inflection is a change in concavity). |
n - 2 |
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A product statement (aka remainder polynomial) is of the form |
f( x ) = d( x ) × q( x ) + r( x )
d( x ) : divisor q( x ) : quotient r(x ) : remainder |
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Remainder Theorem |
If 𝒇 is a polynomial and 𝒇 is divided by (x - a), then the remainder is f( a ). |
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Factor Theorem |
If 𝒇 is a polynomial function, then (x - a) is a factor of 𝒇 ⇔ f( a ) = 0. |
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You can find the output of a function 𝒇 at x = c using the remainder r by doing as follows: |
r = f( c ) |
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The possible rational roots of a polynomial function with integral coefficients can be found by |
factor of constant ------------------------------ factor of leading coef.
aka p/q |
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In general, given a complex number a + bi, then there exists another complex number |
a - bi, called the conjugate. |
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Patterns of i |
i = i i² = -1 i³ = -i i⁴ = 1 |
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Absolute value/ modulus of a complex number a + bi |
▫︎▫︎________ √ a² + b² |
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When graphing a complex number, |
the vertical axis is the imaginary axis and the horizontal axis is the real axis |
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Rational function |
of the form g( x ) / h( x ) where g( x ) and h( x ) are polynomial functions and h( x ) ≠ 0 |
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Vertical asymptote of a rational function |
set denominator = 0 and solve for x |
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Horizontal asymptote of a rational function where m is the degree of the numerator and n is the degree of the denominator
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m = n → HA: y = a/p
m < n → HA: y = 0
m > n → NO HA |
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Slant asymptote |
y = q( x ) where q( x ) is the quotient when the numerator of f( x ) is divided by the denominator |
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Holes of rational functions |
If one or more factors of the function cancel, use the resembles equation to find ordered pair/s for any hole/s
for example: f(x)= [(x+a)(x+b)] / (x+b) → the resembles EQ is g(x) = x+a, and the ordered pair for the hole is (b, g(b)) |
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Limits of rational functions |
Divide the numerator and denominator by x raised to the highest power in the denominator, like so: |
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How to solve polynomial inequalities with a degree > 2 |
1. Write the polynomial so the leading coef. is positive and on the left hand side, and so the inequality is set to 0. 2. Factor the polynomial. 3. Identify critical numbers and region test.
note: same general steps can also be used for rational inequalities |
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A function is one-to-one ⇔ |
its inverse is also a function.
This means each input has exactly one output and each output came from exactly one input. In other words, if f(a) = f(b), then a = b. Graphically, it must pass both the vertical line test and the horizontal line test. m |
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aᵐ × aⁿ = |
aᵐ⁺ⁿ |
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aᵐ/aⁿ = |
aᵐ⁻ⁿ |
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a⁻ᵐ = |
1/aᵐ |
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aᵐ/ⁿ = |
▫︎▫︎▫︎▫︎▫︎______ ⁿ√aᵐ |
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x = aʸ ⇔ |
y = logₐx |
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Properties of logarithms |
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Change of base formula |
▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎ logv logₐv = -------- ▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎ loga |
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Compound interest formulas |
Set time: A = P(1+r/n)ⁿᵗ
Continuously: A = Peʳᵗ
Where P = principal/starting amount, r = rate (percent in decimal form), t = time (in years), and n = number of compounds (yearly) |
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Exponential law of growth or decay |
A(t) = A₀eᵏᵗ
Where A₀ is the initial amount when t (time) = 0 and k is a constant that represents the rate of growth or decay
Growth: k > 0 Decay: k < 0 |
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For an arc length s, with radius r and angle θ (in radians), the arc length s can be found with the formula |
s = rθ |
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For an arc length s, with radius r and angle θ (in radians), the area 𝑨 of the arc can be found using the formula |
𝑨 = ½θr² |
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The three Pythagorean Identities (of trigonometry) of an angle, θ, are |
cos²θ + sin²θ = 1 tan²θ + 1 = sec²θ cot²θ + 1 = csc²θ |
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Sine Half-Angle Identity |
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Cosine Half-Angle Identity |
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Tangent Half-Angle Identity |
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Sine Double Angle Identity |
sin2θ = 2sinθcosθ |
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Cosine Double Angle Identities |
cos2θ = cos²θ - sin²θ cos2θ = 1 - 2sin²θ cos2θ = 2cos²θ - 1 |
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Tangent Double Angle Identity |
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Sum & Difference Angle Identities |
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Heron's Formula for the area of a triangle |
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Sine formulas for the area of a triangle |
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Law of Cosines |
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To convert rectangular coordinates (x,y) into polar coordinates (r,θ) , let r = ? |
▫▫▫▫▫▫_______ r = √ x² + y² |
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To convert rectangular coordinates (x,y) into polar coordinates (r,θ) , let θ = ? |
θ = tan⁻¹ (y ⁄ x) |
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How to multiply & divide complex numbers in polar form |
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