TOPIC 2: Functions

how can degrees (x^y) affect no. real solutions

1 -> max. roots = 1 (not 0) 2 -> max. roots = 2 3 -> max. roots = 3 (not 0) 4 -> max. roots = 4 -> odd numbers cannot have 0 solutions!

The amount of radioactive material, M, in grams, is modeled according to the function ?????? M(t)=250e^−kt, where k>0k>0 and t is time measured in years. It is determined that after 20 years, the amount of radioactive material present is 50 grams.

1) Euler's number means you need to use natural logarithm 2) find k first -> M(20) = 250e^-k(20) -> 50g = 250e^-k(20) -> move all constants to one side -> 50/250 = e^-k(20) -> simplify -> 1/5 = e^-k(20) -> -20k = -ln(5) -> k = 1/20 x ln(5) k = 0.0804 METHOD????

A cup of boiling water is set out to cool. the temperature can be modelled by the function T(t) = A + Be^-0.05t. temp of boiling water is 100C, temp after 10 minutes is 70C A and B are constants, find the value of room temp.

1) SIMULTANEOUS EQUATIONS 2) write what you know -> T(0) = A + B, T(0) = 100C -> T(2) = A + Be^-0.05(10) 3) use Plysmlt2 in GDC to solve simultaneous equation answer: A = 23.755, B = 76.244

Miriam has 80m length of lace that she wants to add to the sides of her rectangular blue blanket what is the maximum area of her blanket?

1) create equation for the area to be put into the GDC 2) find an expression for either x and y and sub. into equation found 3) expand the equation and put into GCD 4) find the y-coordinate of the graph's vertex and THATS THE ANSWER! answer: 400cm^2

area of rectangle is 60cm^2. If it has a width of x cm, which expression represents its perimeter?

1) find an equation for the area -> 60cm^2 = x x y 2) find an expression for y -> y = 60cm^2/x 3) make an equation for the perimeter P = 2x + 2y 4) sub. the equation for y into this equation (P) 5) expand and simplify this expression for the answer answer: 2x + 120cm^2/x

Hugo deposits $900 into an account that pays 4.55% interest per annum compounded monthly. how long will it take Hugo to triple his money?

1) find the compound interest formula 2) set in all the values into the formula, and change nt into x to calculate monthly instalments. 3) 3($900) = $900 (1 + 4.55/100(12))^x 4) 3 = 1 + 4.55/$900^x or 3 = 1.00379^x 5) plug equation into GDC, and set an intercept line to 3 6) y-coordinate of intercept is no. monthly instalments answer: 291 monthly instalment and 24.5 years

given that f(x) = x^2 + 4x - 5, find domain of y = 1/f(x)

1) find the range of the O.G function 2) equate x to 0 3) place equation into quadratic function to get the roots 4) the roots are undefined values, meaning they are equal to the range of the inverse function

find range of function f(x) = 2 + 1/(x^2-1), x ≠ 1

1) find which output values for when x = 1 2) using set notation write the range 3) (-∞,1] U (2, ∞)

Given that f(x) = -4 ∣x + 3∣ - 7, find the range of the function f ????

1) here, the brackets indicate absolute value, meaning that no matter the sign, the bracket is positive (multiple is negative) 2) how to make the bracket equal to 0? Equate x to -3, and solve for f(-3) 3) this should give -4(0) -7, or simply f(-3) = -7 incorrect method??

Annuity definition

finite series of equal payments that occur at regular intervals. once you see monthly/ yearly or quarterly withdraw or deposition of money THINK ANNUITY.

asymptotes -> mathematical notation

horizontal asymptote As x --> ∞, f(x) --> k vertical asymptote as x --> h, f(x) --> ∞

The half-life of a radioactive material is approx. 5750 years. if initial amount is 13 grams, how much will remain after 13500 years?

1) start by finding decay factor -> in this case, the mass of radioactive material is constantly halved, meaning decay factor is 1/2 2) create equation and plug into GDC -> A0 x (1/2)^k (k = no. halv lives) 3) how many halv-lives fit into 13500 years? divide 13500 by 5750, and set that value as the exponent. 4) plug in all known values into new equation and solve -> At = 13g (1/2)^2.348 answer: 2.55g

The points A(1,8) and B(5,6) lie on a line. the line 6x + by + 5 = 0 is perpendicular to this line. what is the value of b

1) using the provided points, we get the gradient of the original line 2) flip this gradient to make it the gradient of perpendicular line (should be 2) 3) return to equation of perpendicular line and rearrange ti get y alone on one side. 4) the expression for gradient is then 6x/b, and as this must equal 2, solve for b answer: b = -3x

practice problem ????? find the domain of function f(x) = (2 + 3x)/(3-2x). Function has range (-∞, 3) U (3, ∞)

1) view range, spot undefined values 2) undefined value is 3, meaning 3 = (2 + 3x)/(3-2x) does not exist 3)

TOP TIP: how to count no. VERTICAL asymptotes from GDC

1) write function into calculator 2) enter table of values 3) count no. errors

annuity example problem Gary borrows 1,000,000 and plans to repay it over 20 years. the interest rate is 2,3% per annum compounded quarterly. Find the quarterly amount he must pay.

1) write up all known/unknown values PV = 1,000,000 PMT = ? r = 2.3 n = 4 x 20 = 80 (this is measured in quarters) C/y = 4 P/y = 4 place all values into the TMV solver and calculate PMT answer: 15629.92 USD

identify the range of function y = ln(x^6)

ALL LOGARITHMIC FUCNTIONS HAVE UNLIMITED RANGE!! 1) as per statement above, y can be all real numbers

which is NOT a polynomial function? a) f(x) = 2x^3(3x + 4) b) g(x) = -x(x^3-2) c) h(x) = 5√x + 2

C - exponent of variable √x is NOT an integer -> x^1/2

the interest rate per annum is 6.2%, what is the quarterly, and semi-annual interest?

DIVIDE BY EITHER 4 OR 2.

second term of geometric sequence is 3, and fifth term is 15, what is the common ratio?

IMPORTANT: 3 x r^3 = U5 or 15 this way, it is much easier to calculate common ratio without the first or consecutive terms.

Annuity Formula

PV = p x (1 - (1 +r)^-n/r

'normal' definition

Same as perpendicular line. unknown constant + set of points: simply flip the gradient and use a set of points to calculate the constant/y-intercept. unknown gradient + set of points plot the set of points into the formula and calculate the gradient.

Rosetta takes out a loan of 25000 EUR25000 EUR from a bank. She will pay interest at the rate of 6.2%6.2% per annum, which will compound quarterly. Let Vn represent the value of Rosetta's loan after n compounding periods. What is the best model to represent the value of Rosetta's loan?

The interest rate is per annum but there are 4 compounding periods (n = 4), so the interest rate per period is 6.24 = 1.55 Hence, R = 1 + 1.55100 = 1.0155 So the correct model is given by: Vn = V0 × (R)n -> Vn = 25000 × (1.0155)n IT IS NOT RAISED TO 4N, AS THAT IS ALEADY ACCOUNTED FOR BY ALTERING THE INTEREST RATE FROM ANNUAL TO QUARTERLY

regression lines

The line of best fit drawn through a scatterplot how to find on a calculator: stat -> edit -> place values into table -> stat -> calc. -> chose regression line -> press enter

direct variation

The relationship between two variable quantities that have a constant ratio. e.g: linear equations relate two TWO variables, x and y that are in direct variation, where m is the constant of proportionality

Discriminant example problem 4x^2 + bx + 1 = 0 - find b using the quadratic formula

USE discriminant equation √b^2-4ac - place known values into the equation and find b - set equal to zero?

Form of a geometric growth function..

Vn=V0×(R)n Vo = initial value R = interest rate n = no. compounding periods

Asymptote

a line that continually approaches a given curve but does not meet it at any finite distance.

Domain

all possible input values of a function

range

all possible output values

Inverse Variation

an equation of the form y = m/x or m = xy, where k is not equal to zero - as one variable increases, the other decreases at a constant rate (m) given by the constant of proportionality. e.g: when driving a car, the fater you go, the less travel time there is.

Discriminant

b^2-4ac - (part of quadratic formula) -if positive, equation has 2 distinct real solutions, if 0 then 1 real solution, if negative there are no solutions

find the derivative of 2x^2 - 3/x????

derivative of quotient???

Horizontal Asymptote

describes the value of y as x increases without bound! as x --> + infinity, graph tends toward horizontal asymptote y = k as x --> - infinity, graph tends toward horizontal asymptote y = k in the form y = a/(x-h) + k, horizontal asymptote is k (a, which is added to k, is divided by higher and higher values, meaning it approaches the true value of k more and more)

extrapolation vs interpolation

extrapolation is an estimation of a value based on extending a known sequence of values or facts beyond the area that is certainly known. interpolation is an estimation of a value within two known values in a sequence of values.

Derivative of a function

f'(x) = lim h->0 f(x+h) - f(x)/h when taking the derivative, you lessen the max degree of a function by 1. e.g: quadratic function becomes linear function method 1) multiply coefficient by O.G exponent 2) subtract 1 from exponent 3) (this is done individually for each expression in the equation)

practice problem ????? function f(x) = (2 - x)/(2 + x). Function has range (-∞, a) U (a, ∞), find value of a

f(x)=2−x2+x=a is not solvable 2−x=a(2+x) is not solvable 2−x=2a+ax is not solvable

exponential functions RULES..

if a>0 - domain is -∞, +∞ if k>0 -> range is (c,∞) if k<0 -> range is (c,-∞) - y-intercept is (0, k + c) If a > 1 and k > 0 - function is growing - the y = c is a horizontal asymptote (f(x) --> c asymptotically) If 0<a and k>0 - function is decaying - the y = c is a horizontal asymptote (f(x) --> c asymptotically)

turning points of functions

max./min. points are called EXTREME global max/min -> largest/smallest of whole graph local max/min -> largest/smallest comparing to surrounding values

direct variation as nth power

one variable can also vary to the power of another e.g: area of the circle is given by -> A = πr^2 where A is in direct variation with the square of the radius, and π is the constant of proportionality

how to reflect exponential graph across y-axis?

reverse the sign of the coefficient! a^x becomes -a^x

quadratic function

standard form -> y = ax^2 + bx + c y = output a = gradient/concavity of graph b = horizontal placement of parabola (line of symmetry) x = input c = vertical placement/constant/y-intercept

linear function

standard form -> y = mx+c y = output m = gradient x = input c = constant/y-intercept - relates two TWO variables, x and y that are in direct variation!

exponential function

standard form: Ka^x + c k: can equal any number, except 0 a: determines steepness c + k: y-intercept

polynomial functions

standard form: f(x) = a(n)x^n....a2x^2 + a1x + a(0) -a(n) is leading coefficient -x^n is the degree -a(n)x^n is leading term (has highest degree) a - determines the steepness no. terms - determine whether graph is parabola or curvy line, and the more terms, the steeper the line 1) if the leading term is above 4, and odd, line is curvy 2) if leading term is above 4 and even, line is parabola 3) if leading term is below 4, curvy line - see discriminates

cubic

standard form: f(x) = ax^3 + bx^2 + ex + d a - determines concavity (affects max. points) b - determines horizontal placement (--//--) e - affects max. points (increase in e, meaning larger/smaller max/min points. - when a and b = 0, e is the gradient of the straight line d - y-intercept cubic functions are polynomials to the third degree

Power functions

standard form: f(x) = ax^n - a and n are real numbers, and a is the coefficient n: determines shape of graph (if odd, curvy, if even, parabola) - with even exponent, the signs are irrelevant, and the graph cannot have negative values, NOT with odd even: as x --> +/- ∞, f(x) --> ∞ odd: as x --> + ∞, f(x) --> + ∞ and as x --> -∞, f(x) = -∞

natural exponential function

standard form: f(x)=e^x - e = Euler''s number (2.71828 - irrational) e^x = exponential growth e^-x = exponential decay ((1/e)^x)

the derivative of a third degree/cubic function shows...

the growth rate. the derivative is a parabola, where the vertex is equal to the highest growth rate in the original function.

If two points are COLINEAR what does this mean?

they lie on the same line

How to calculate investment growth using inflation?

use the standard compound interest formula, but divide by the percentage of inflation eg. investment of $100 grows by 8.1%, yearly compounded if the inflation per year is 3.2%, what is the yield after 5 years? equation becomes: FV = $100 x (1.081/1.032)^5

vertical asymptote

vertical line that describes behavior of a graph y=f(x) as x tends to the value where f(x) is NOT DEFINED. describes the behavior of x as f(x)/y increases WITHOUT BOUND. How to find: 1) set the DENOMINATOR of equation y = a/(x-h) + k to 0 - done to find values where x is NOT defined, aka the vertical asymptote. e.g y = 1/(x-1) -1 horizontal asymptote = -1 vertical asymptote = 1 (1-1 = 0, simply reverse the sign of constant)

how to solve annuities on GDC?

when questions mention regular payments, think ANNUITIES place the value of yearly, monthly or quarterly payment into PMT and calculate any values required!

vertex form

y=a(x-h)^2+k where (h, k) is the vertex of the parabola