Functions: Classifications, Rules, Solving
Vertical Asymptotes - how do you find?
1.Factor out the function if needed and simplify as much as possible (removing common factors) 2.Determine the zeros for the numerator and for the denominator 3.Look for the value(s) of x where the denominator = 0 but the numerator ≠ 0 a.If there are multiple x=0's, ensure that you only select the one that makes this statement true (in ex. Above x=1 makes the function undefined, so both the denominator and numerator = 0, this is not an answer. Only x = -3 is correct because it satisfies both parameters.) If the denominator can never = 0, then the line has no vertical asymptotes
How would you determine the difference between slopes?
2 perpendicular lines have slopes of m and (-1/m), their difference is m-(-1/m) or m+ (1/m), to determine if the difference between slopes is possible of the given answer set, set the answer choice equal to this equation and see if the m value resulting from the quadratic formula to solve yields a proper answer (improper answers would include things like a negative root in quadratic formula.)
quadratic function
A function in which the greatest power of the variable is 2. y=ax^2+bx+c. Domain = all real #s, Range = all real #s except those that make the equation = 0. Can find the zeros by solving using the quadratic formula, shown. If has just 1 zero, it touches the x-axis once; if 2 roots, it crosses the x-axis; if no roots, it doesn't contact the x-axis.
transcendental function
A function that cannot be expressed in terms of algebraic operations, such as an exponential or logarithmic function.
How do you find the horizontal asymptote?
Compare the degrees in the numerator vs denominator: n < d asymptote @ y=0 n = d asymptote @ y = n/d n > d no horizontal asymptote
What is the greatest number of complex roots a 17th degree polynomial can have?
Complex solutions always come in pairs. Therefore, the number of possible complex solutions is the greatest even number equal to or less than the power of the polynomial. A 17th degree polynomial can have at most 16 complex roots.
The Greatest Integer Function
D: (-∞,+∞) R: (-∞,+∞) denoted by y = [x]. less than or equal to x. In essence, it rounds down a real number to the nearest integer
Domain and Range of Square Root Function
D: [0,∞) R: [0,∞) CANNOT be NEG. #s b/c square root is never negative.
How would you find the domain of a polynomial function? f(x)= x^2 + 3x + 1 or f(x) = 2x + 1
Domain = all real #s
How would you find the domain of a root on top function? f(x) = [root (x+1)] / (x^2 -4)
Domain = denominator cannot = 0 AND value under root must be greater than or equal to 0, BOTH of these restrictions are in your restrictions for this domain.
How would you find the domain of a fraction function? f(x) = (2x + 1) / (x^2 + 3x + 1)
Domain = set denominator to not equal 0, anything that makes = 0 will create an undefined function and this is a domain restriction
How would you find the domain of a root in the denominator function? f(x) = (x^2 + 2x) / root (x+1)
Domain = set par under the root > 0 to solve for domain restrictions
How would you find the domain of a root function? f(x) = sq root (x^2 + 2x + 1)
Domain = set part under root greater than or equal to 0 to find domain restrictions Square root functions are defined for all real numbers except those which result in a negative expression below the square root. When setting up the domain restriction WATCH for NEGATIVES UNDER ROOT, this will FLIP the sign!
Domain and Range of Absolute Value Function
Domain: (−∞, ∞) Range: [0, ∞) Domain CAN be negative b/c absolute values can change any value to a positive result.
Translations: f(bx)
Horizontal COMPRESSION if |b| > 1 STRETCH 0 < |b| < 1 (x, y) --> (x/b, y)
Translating Graph: What does '+2' do? f(x) = (x + 2)^2
Horizontal Shift of x^2 graph to the LEFT (negative shift with a positive number when horizontal, in ( )'s)
Translating Graph: What does '-2' do? f(x) = (x - 2)^2
Horizontal Shift of x^2 graph to the RIGHT (positive shift with a negative number when horizontal, in ( )'s)
Translating Graph: What does 'k' do? f(x) = y = sin (x + k)
Horizontal Shift: similar to other transformation graphs, if k is positive, shift LEFT. If k is negative, shift RIGHT. (ie: y = sin (x + pi/2) means shift left by 1/2 pi)
Translating Graph: What does 'b' do? f(x) = y = sin bx
Horizontal stretch, impacts the PERIOD. If positive, means that the function will rise and fall at that many times the normal rate (ie: y = sin 2x will rise and fall at 2 times the normal rate) Period becomes 2pi/b
f(b*x) results in what kind of stretch / compression? How does it stretch? How would it compress?
Horizontal: |a| > 1 = horizontal compression 0 < |a| < 1 horizontal stretch
Translations: -f(x)
Reflection over x-axis (x, y) --> (x, -y)
Translations: f(-x)
Reflection over y-axis (x, y) --> (-x, y)
perpendicular lines -- how do you determine points that could be on the same line?
Take the points given and find the slope, m. Solve for b in y=mx+b formula. Then the other line would have a negative inverse slope (-1/m) and the same b value. Test points using this slope OR test points to see if they have this inverse slope value.
What happens when the numerator =0
The solution to the problem will = 0 WHEN ASKED TO FIND THE 0s OF A COMPOUND FORMULA, ONLY SOLVE FOR WHAT WILL MAKE THE NUMERATOR = 0; the denominator =0 would mean undefined, though these are the 'zeros of the function' where x=the denominator values of 0.
Double Angle Identities
Using double the angle of a trig function to reduce into other trig functions that are not double angles in order to solve/prove angle identities. 1.Use trig identities (tan/sin/cos, etc) to make the reference triangle. 2.Solve for the missing side using Pythagorean theorem. 3.Based on the points given for the triangle, fill in the trig identities needed to solve for. 4.Solve, plugging in the values. Tan can also = (sin 2theta/ cos 2 theta) instead!
Half Angle Identities
Using half the angle of a trig function to reduce into other trig functions based on their properties to solve/prove angle identities. 1.Use trig identities (tan/sin/cos, etc) to make the reference triangle. 2.Solve for the missing side using Pythagorean theorem. 3.Based on the points given for the triangle, fill in the trig identities needed to solve for. 4.Solve, plugging in the values. **NOTE: The +/- symbols aren't because it can be a + or - --> they are to indicate you need to choose which it is based on what quadrant the angle is falling into and the properties of that trig identity in that quadrant. *NOTE: you can use your reciprocal identities for trig fn.s to solve for the identities not listed below. Ie: inverse of tan is cot, so you'd just flip the tan formula over. Sec = 1/cos, so you'd just put 1 over to cos formula. *NOTE: even though you may be adjusting to half angles to find on unit circle, USE THE SYMBOL OF THE QUADRANT THE ACTUAL Angle is being measured for NOT the sign / quadrant of the half angle you're using to solve!
Domain Restrictions
Values that would make the function (or denominator of a function if in a fraction) = 0 = undefined. These MUST be done BEFORE factoring out like terms!
Translations: a(f(x))
Vertical STRETCH if |a| > 1 COMPRESSION 0 < |a| <1 (x, y) --> (x, ay)
Translating Graph: What does 'k' do? f(x) = y = sin x + k
Vertical Shift: shift up if + or down if - (ie: y = sin x + 1)
Translating Graph: What does '-2' do? f(x) = x^2 - 2
Vertical shift of graph of x^2 DOWN 2 units
Translating Graph: What does '+2' do? f(x) = x^2 + 2
Vertical shift of graph of x^2 UP 2 units
Translating Graph: What does 'a' do? f(x) = y = a sin x
Vertical stretch to sin x function. Becomes the new AMPLITUDE (if y= 2 sin x, the graph will be cycling between 2 & -2 for its peaks and pits)
a*f(x) results in what kind of stretch / compression? How does it stretch? How would it compress?
Vertical: |a| > 1 = vertical stretch 0 < |a| < 1 vertical compression
polynomial function
a continuous function that can be described by a polynomial equation (can have multiple terms and multiple powers of x) in one variable. Domain = all real #s, if highest degree (power) is even, then the range is the set of all real numbers that satisfy the function; if odd, the range is all real #s.
linear function
a function in which the graph of the solutions forms a line. y=mx+b and x=(-b/m) for the x-zero of the line.
Translating Graph: What does 'b' do? f(x) = (bx)^2
a is in the ( )'s as a coefficient of x being squared together |b| > 1 compress horizontally 0< |b| < 1 stretch horizontally
Translating Graph: What does 'a' do? f(x) = a(x^2)
a is outside of the ( )'s as a coefficient of x squared |a| > 1 stretch vertically 0< |a| < 1 compress vertically
constant function
a linear function of the form f(x)=b, where b is a real number. No independent variable so the function has a constant value for all x, the graph is a horizontal line
rate of change formula -- used in rate of change over an interval: ie rate of change of f(x)=√(x-1) over [5, 17]
change in value/time f(x) = (f(b) - f(a)) / b-a f(x)=(√(17-1)-√(5-1))/(17-5) = 1/6
Polynomial end point behavior: if a 0 has odd multiplicity (is an odd exponent), _______ at 0
crosses x-axis at 0
Polynomial end point behavior: leading degree is odd and coefficient > 0
falls left / rises right
Polynomial end point behavior: exponent even and coefficient <0
falls left AND right
monotone function
function whose graph either constantly increases or decreases
algebraic function
functions that exclusively use polynomials and roots. Can be joined by any operation but cannot include variables as exponents.
odd function
graph is symmetrical with respect to the origin; f(-x)=-f(x) f(x)=x^3 and f(x) =ax^n where n = positive ODD integer
even function
graph is symmetrical with respect to the y-axis; f(x) = f(-x)
Translations: f(x + h)
horizontal translation LEFT h units (Add to x= "go west" move left if adding an h to x)
Translations: f(x - h)
horizontal translation RIGHT h units
Y-Z plane
plane that comes off of the y-axis into 3D forward/ back; any point in this plane will have an x = 0. Same works for X-Z Plane y=0. Plug in and solve for the other variables to find variable intersections.
Translating Graph: What does a negative do? f(x) = -x^2
reflect the graph across the x-axis. OR inverts the graph, like when have x^3
Translating Graph: What does 'a' do? f(x) = y = -a sin x
reflection across x axis since a is negative & vertical stretch / compression depending on a's value: |a| > 1 stretch vertically 0< |a| < 1 compress vertically Amplitude is still |a|.
Polynomial end point behavior: exponent even and coefficient >0
rises left AND right
Polynomial end point behavior: exponent is odd and coefficient <0
rises left, falls right
Sum and Difference Formulas
sin[x(+/-)y]=sin(x)cos(y)[+/-]cos(x)sin(y) cos[x(+/-)y]=cos(x)cos(y)[-/+]sin(x)sin(y) ----notice that for cos, if it is positive in the first part, is negative in the second part tan[x(+/-)y]=tan(x)[+/-]tan(y)/1[-/+]tan(x)tan(y) ---notice that the bottoms are opposite the top signs for Tan
how do you solve an inverse function f-1(x)?
swap the input and output of the function values to 'undo' the original function. (x, y) --> (y, x)
In finding the vertical asymptotes, what happens if an answer makes the numerator and denominator = 0?
the numerator cannot = 0, only finding what will make just the denominator = 0. If one of the solutions would make the numerator = 0 too, this would create an undefined solution, and this x root would not be a possible answer.
What happens when the denominator =0
the solution to the problem will = UNDEFINED These are EXCLUDED Values or Domain Restrictions. You need to find ALL of these (so if canceled out a like factor, still need to pull through this domain restriction--ex: (x-3)(x+3) would be +/- 3)
Polynomial end point behavior: if a 0 has even multiplicity (is an even exponent), _______ at 0
touches but doesn't cross x-axis at 0
Translations: f(x) - d
vertical translation DOWN d units
Translations: f(x) + d
vertical translation UP d units Add to y "go high" (move up if adding a d)
what is point slope form if given a point and slope?
y-y1 = m(x-x1)
When would a log be reflected about the x-axis?
y= - a logb(x ± c) - d When a negative comes before the function, reflect about the x-axis
When would a log be reflected about the y-axis?
y= a logb(- (x ± c)) - d When a negative comes before the x variable, reflect about the y-axis
Where can you determine if there's a stretch or compression on a log function?
y= a logb(x ± c) - d When multiplied by a constant, a > 0 = VERTICAL stretch or compression. a > 1 = stretch // a < 1 = compression
Where can you determine if there's a horizontal shift on a log function?
y= logb(x ± c) When a constant c is added to the input of the parent function, the result is a horizontal shift c units in the opposite direction of the sign on c. + = shift left (c > 0), - = shift right (c < 0).
Where can you determine if there's a vertical shift on a log function?
y= logb(x ± c) - d When a constant d is added, a vertical shift happens for +d up (d>0), or -d down (d<0).
parabola formula (vertex form)
y=a(x-h)^2+k, (h,k)=vertex, intercept/width & direction
parabola formula (standard form)
y=ax^2 + bx + c vertex ( (-b/2a), f(-b/2a))
Identity Function Graph
y=x, or where every value of y = a corresponding value of x with the zeros at (0, 0) and slope = 1