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One-to-one function - correct answers ✅each x value pairs to exactly one UNIQUE y value
- Passes HLT & VLT (No repeated x or y values in a table) Function - correct answers ✅each x value pairs to exactly one y value
- Passes VLT (No repeated x values in a table) Degree of a polynomial - correct answers ✅the greatest degree of any term in the polynomial Leading term - correct answers ✅the term with the highest degree What is the maximum number of turning points a polynomial can have? - correct answers ✅n- n = the degree of the polynomial Inflection point - correct answers ✅= Leading exponent
- 2
Point where concavity changes (CU to CD or CD or CU) Asymptote - correct answers ✅a line that a graph approaches but never crosses Asymptote(s) of an exponential function - correct answers ✅horizontal Asymptote(s) of a logarithmic function - correct answers ✅vertical Vertical stretch - correct answers ✅the whole function, f(x), is multiplied by a (where a > 1) Vertical compression - correct answers ✅the whole function, f(x) is multiplied by a (where a is a fraction: 0 < a < 1) Horizontal stretch - correct answers ✅- x is multiplied by a (where a is a fraction: 0 < a < 1)
- graph stretches away from the y-axis Horizontal compression - correct answers ✅- x is multiplied by a (where a > 1)
A function is symmetric over the origin if - correct answers ✅f(-x) = -f(x)
- It is an odd function f^-1(f(x)) - correct answers ✅x f(n) = m, f^-1(m) = ___ - correct answers ✅n leading coefficient - correct answers ✅The coefficient of the term with the highest degree (leading term) Solve an inequality with an absolute value - correct answers ✅1. Isolate the absolute value
- Solve for a + other side
- Solve for a - other side AND flip the inequality Complex zeroes - correct answers ✅When solving for all the zeroes, there will be a negative under the square root. Replace it with i
- Its conjugate is ALSO a zero Number of zeros of a polynomial - correct answers ✅= the exponent of the leading term (the highest exponent)
- some can be imaginary and unseen on the graph
Complex conjugate - correct answers ✅Complex numbers: a+bi and a-bi
- If a function has a zero at a+bi, it ALSO has a zero at a- bi Multiplicity of a zero - correct answers ✅Number of times a zero's factor occurs in a polynomial
- If odd, the line passes through that zero
- If even, the line will be tangent to the x-axis (bounce off) at that zero End behavior (EB) - correct answers ✅The behavior as x approaches positive or negative infinity:
- EB of an even function is the same for its -∞ & ∞
- EB of an odd function is the opposite for its -∞ & ∞ How to find the equation of a quadratic from a table of values? - correct answers ✅- AOS = avg of x values from equal y values
- OR plug two/three points into ax^2 + bx + c form & solve system of equations (a is 2nd difference/2) Linear, quadratic, or exponential from table of values? - correct answers ✅- Linear = First difference in y is constant
coefficients, then every rational zero of f(x) takes the form:
p/q = factor of a (constant) / factor of a ⌄n (leading
coefficient) Rational function - correct answers ✅R(x) = f(x)/g(x) f(x) is a polynomial or monomial/constant g(x) is a polynomial & not 0 End behavior of a rational function - correct answers ✅1. Numerator degree > denominator degree:
- EB matches the EB of the quotient of the leading terms
- Numerator degree = denominator degree:
- EB approaches the horizontal asymptote (= ratio of leading terms) in both directions
- Numerator degree < denominator degree:
- EB approaches the horizontal asymptote y = 0 in both directions Types of Discontinuity - correct answers ✅1. Infinite/asymptotic
- Removable/point
- Jump
AP Exam: Round to ___ decimal places - correct answers ✅ 3 THE THIRD ONE! AP exam, you should round to _____ decimal places - correct answers ✅ 3 e = - correct answers ✅roughly 2. A horizontal translation of an exponential function is... - correct answers ✅the same as giving it a vertical stretch or changing its vertical stretch E.g., if f(x) = b^x, then f(x+1) = b^(x+1) = b^x * b (in this case, b becomes a, the vertical stretch) A horizontal stretch/compression of an exponential function is... - correct answers ✅the same as changing its base E.g., f(x) = b^x, then f(3x) = b^3x = (b^3)^x (effectively changing the base from b to b^3) A reflection over the y axis of an exponential function is...
- correct answers ✅the same as making the base its inverse
log (x^P) = - correct answers ✅P * log (x) log (3^x) = - correct answers ✅x * log (3) log without a base on a calculator is - correct answers ✅log (base 10) To take the natural log on a calculator? - correct answers ✅ln log(base e) = - correct answers ✅ln if log(base a)x = log(base a)y, then - correct answers ✅x = y if ln(x) = ln(y), then - correct answers ✅x = y log(base a)1 = - correct answers ✅0, because a^0 = 1 log(base a)a = - correct answers ✅1, because a^1 = a log(base a)a^x = - correct answers ✅x, because a^x = a^x
log(base x) of ≤ 0 - correct answers ✅does not exist (gives a calculation error) Natural vs. common log - correct answers ✅Natural = base e Common = base 10 The change of base formula - correct answers ✅log(base b)x = log(x)/log(b) The same is true for any base on the right side as well, including e (the natural log (ln)) How to find the domain/range of the inverse of a function
- correct answers ✅The range of the inverse function is the domain of the original The domain of the inverse is the range of the original What is one (slow but notable) way to find a log function (or inverse function - correct answers ✅Create a table of values for the original function (the exponential version), then swap the x and y values to get the inverse (log) function's table of values
- Logarithmic equations (with logs) How to solve a logarithmic equation with a constant (steps) - correct answers ✅1. Bring logs with the same base to the same side, and combine into 1 log
- Get into the form: log(base b)c = a
- Rewrite in exponential form: b^a = c and solve
- Check the solution(s) in the original equation (extraneous if it creates a log of zero or a negative number) If you're modeling a function with continuous exponential or logarithmic change... - correct answers ✅The base is e. (For logs, this means use ln). e^ln(x)= - correct answers ✅x for x> Continuous exponential change formula - correct answers ✅f(x) = A(e)^rt A = initial amount r = rate of change per unit of time (e.g., if +1%, it's 1.01) t = time
When you have to make an exponential function for something in nature (e.g., bacterial growth)... - correct answers ✅USE BASE e e - correct answers ✅2. Residual - correct answers ✅observed value - predicted value Residual plot - correct answers ✅x-axis = independent variable y-axis = residual values (difference between observed and predicted values)
- If points are randomly dispersed, the model is appropriate Arithmetic sequence - correct answers ✅The difference between consecutive terms is constant (called the common difference) Term of arithmetic sequence formula - correct answers ✅An = Ak + d(n - k) An = the nth term (you are trying to find)
Sequences vs. functions - correct answers ✅sequences are discrete; functions are continuous Period (english definition) - correct answers ✅The domain (x-value interval) required to complete a full cycle of a periodic function Period (mathematic condition) - correct answers ✅A period must be the smallest value K such that f(x+K) = f(x)
- The interval it takes to return to the same value
- Necessary but not sufficient to be a period Cycle - correct answers ✅A complete pattern of y values in a periodic function Periodic function - correct answers ✅A function that repeats a pattern of y-values (cycle) at regular intervals (periods) tan(45) = - correct answers ✅= x/x = 1 Because it means the triangle is right isosceles, so the legs are equal, yielding x/x
Trig functions are ______ functions - correct answers ✅periodic Standard position - correct answers ✅An angle with its vertex at the origin and its initial side fixed on the positive x-axis Initial vs. terminal side - correct answers ✅Initial: The fixed side that does not rotate Terminal: The non-fixed side that rotates Terminal side rotation - correct answers ✅- If it rotates clockwise (starting into negative y values), the angle is negative
- If it rotates counterclockwise (starting into positive y values) the angle is positive Vertex of an angle - correct answers ✅The common endpoint of the two lines that form an angle A 1 radian center angle in a circle means... - correct answers ✅the 1 radian angle's arc length equals the radius Coterminal angles - correct answers ✅Angles in standard position that share a terminal side
cosine x, sine y... (repeat for memorization) On the unit circle, tan θ = ___ - correct answers ✅tan θ = m = y/x = sin θ/cos θ where the terminal side and the unit circle intersect at (x, y) because the hypotenuse (radius) = 1 Formula for finding point P using the radius and angle - correct answers ✅(r * cosθ, r* sinθ) Formula for the slope of a terminal ray - correct answers ✅sinθ/cosθ (which also equals tanθ) sin(π/4) - correct answers ✅√2/ sin(π/3) - correct answers ✅√3/ sin(π/6) - correct answers ✅1/ cos(π/4) - correct answers ✅√2/ cos(π/3) - correct answers ✅1/
cos(π/6) - correct answers ✅√3/ Reference angle - correct answers ✅The angle the terminal side of any angle makes with the x-axis (rather than the initial side) For a point on a circle, given its angle and radius, x = _______ and y = _______ - correct answers ✅x = r * cosθ y = r* sinθ Sinusoidal function - correct answers ✅A function that is a transformed version of f(x) = sin(x)
- E.g., g(x) = sin(x - pi/2) is a sinusoidal function b/c it's sin(x) with a horizontal shift to the right of pi/ sin(x) graph - correct answers ✅Starts at the origin (0,
- and is initially increasing as x increases cos(x) graph - correct answers ✅Starts at (0, 1) and is initially decreasing as x increases Domain, range, and period of sin(x) and cos(x) - correct answers ✅Domain: all real numbers Range: [-1, 1]
