MHF4U Grade 12 Advanced Functions – Exam

Grade 12 – Advanced Functions

 

Exam

 

Unit 1: Polynomial Functions

 

  • Polynomial Expression has the form:

anxn+an-1xn-1+an-2xn-1+ … + a3x3+ a2x2+ a1x+ a0

    • n: whole number
    • x: variable
    • a: coefficient X ER
    • Degree: the highest exponent on variable x, which is n.
    • Leading Coefficient: anxn
  • Power Functions: y = a*xn , n EI
  • Even degree power functions may have line symmetry along y-axis.
  • Odd degree power functions may have point symmetry about origin.

 

  • Finite Differences: a ( n! )
  • Properties for Odd Functions:
    • Positive Leading Coefficients:
      • Goes from Q3 to Q1
    • Negative Leading Coefficients:
      • Goes from Q2 to Q4
    • Domain {XER} , Range {YER}
  • Properties for Even Functions:
    • Positive Leading Coefficients:
      • Goes from Q2 to Q1
    • Negative Leading Coefficients:
      • Goes from Q3 to Q4
    • Have local maximum/minimum points
    • Domain {XER}, Range above/beyond maximum/minimum points

 

  • Sketching Graphs
    • Sketch graphs according to its degree
    • Polynomial functions have 0 to maximum of n x-intercepts, where n is the degree.
    • Polynomial functions can also have at most n – 1 local max/mins, where n is the degree.
    • X – Intercepts are the roots of the function.
    • Factors of a factored form equation are also the roots.

 

    • Roots can have different orders, and are graphed differently:
      • Order 1 root [ie (x-1) ]
        • passes through the x-intercept
      • Order 2 root [ie x2 ]
        • passes in tangent to the x-intercept
      • Order 3 root [ie x3]
        • passes through intercept like y = x3 at the origin
  • Even Function Test: f(-x) = f(x)
  • Odd Function Test: f(-x) = -f(x)

 

  • Transformations of polynomial functions of the form y = a [ k (x-d) ]n + c :
    • a >= 1 : vertical stretch by a factor of a.
    • 0 <= a <= 1 : vertical compression by factor of a.
    • k >= 1 : horizontal compression by a factor of 1/k
    • 0 <= a <= 1 : horizontal stretch by a factor of 1/k. (be careful to factor out sometimes).
    • d : horizontal translation by d units left or right (be careful to factor out sometimes).
    • c : vertical translation by c units up or down.

 

  • Average Rates of Change:
    • Delta Y / Delta X
    • Slope of secant of 2 points give the average rate of change
  • Instantaneous Rates of Change:
    • To calculate without calculus, substitute a number thats really close to the time wanted.
    • Use that value to achieve a the slope of a tangent at that point
    • Instantaneous rates of change is the tangent at those points

 

Unit 2: Polynomial Equations and Inequalities

  • Remainder Theorem
    • Long division can be used to divide polynomials
    • The division statement can be said to be:

P(x) = (x-b)(Q(x))+R where R is the remainder, x-b is a factor, Q(x) is the divisor

    • Theory states that when P(x) is divided by (x-b), the remainder is P(b), where if P(b) = 0, then b is a root of the equation.
  • Factor Theorem
    • States that (x-b) is a factor of an equation if P(b) = 0
    • (ax – b) is a factor if P(b/a) = 0

 

  • Integral Zero Theorem
    • If (x – b) is a factor, and if the polynomial equation had a leading coefficient of 1, then b must be a factor of the constant term

 

  • Rational Zero Theorem
    • If (ax – b) is a factor, and the polynomial equation had a leading coefficient greater than 1, then x = b/a is a rational zero of P(x), such that
      • b is a factor of the constant term
      • a is a factor of the leading coefficient
      • (ax – b) is a factor

 

  • Families of Polynomial Equations
    • A family is a group of equations with the same charactoristics
    • They have the same x-intercepts
    • They take the form: y = k(x-a)(x-b)(x-c)
    • Families can be converted into an equation with a given point

 

  • Solving Inequalities
    • Put all terms on one side
    • Factor if possible
    • Use Interval Table / graph to satisfy the inequality (areas that are < or > 0)

 

Unit 3: Rational Equations

  • The reciprocal of a linear function has the form:

f(x) = 1 / kx – c

  • The restriction on a domain of a reciprocal linear function can be determined by finding the value of x that makes the denominator equal to zero, that is x = c / k.
  • The Vertical Asymptote of a reciprocal linear function has an equation of the form x = k / c. 
  • The horizontal asymptote of a reciprocal linear function has equation y = 0.
    • If k > 0, the left branch of a reciprocal linear function has a negative, decreasing slope, and the right branch has a negative, increasing slope.
      • Basically occupies Q3 and Q1.
    • If k < 0, the left branch of a reciprocal linear function has a positive, increasing slow, and the right branch has a positive, decreasing slope.
      • Basically occupies Q2 and Q4.

 

  • Rational quadratic functions can be analyzed using key features: asymptotes, intercepts, slope (positive or negative, increasing or decreasing), domain, range, and positive and negative intervals.
  • Reciprocal of quadratic functions with two zeros have three parts, with the middle one reaching a maximum or minimum points. This point is equidistant from the two vertical asymptotes.
  • The behavior near asymptotes is similar to that of reciprocals of linear functions.
  • All of the behaviors listed above can be predicted by analyzing the roots of the quadratic relation to the denominator.

 

  • A rational function of the form f(x) = (ax + b) / (cx + d) has the following key features:
    • The vertical asymptote can be found by setting the denominator equal to zero and solving for x, provided the numerator does not have the same zero.
    • The horizontal asymptote can be found by dividing each term in both the numerator and the denominator by x and investigating the behavior of the function as x -> positive or negative infinity.
    • The coefficient b acts to stretch the curve but has no effect on the asymptotes, domain, or range.
    • The coefficient d shifts the vertical asymptote.
    • The two branches of the graph of the function are equidistant from the point of intersection of the vertical and horizontal asymptotes.
  • Analysis of End Behavior
    • For vertical asymptote
      • Substitute a number very close to the VA from the right, and a number from the left
      • Analyze the result of that number and express the end behavior
      • Whether As x -> VA +/- , y -> +/- infinity
    • For horizontal asymptote
      • Substitute a very large negative and positive number for x and analyze the behavior of y.
      • Express the end behavior with the results from that substitution
        • As x -> +/- Infinity, y -> HA from above/below

 

  • To solve rational equations algebraically, start by factoring the expressions in the numerator and denominator to find asymptotes and restrictions.
  • Next, multiply both sides by the factored denominators, and simplify to obtain a polynomial equation. Then solve.

 

  • For Rational inequalities
    • Set the right side of the equation zero.
    • Factor the expression to find restrictions
    • Based on the assumption that x = a / b is true if and only if a * b = x.
      • On the left side of the equation, take the denominator and multiply it by the numerator.
      • Since the equation is already factored, the roots are clearly shown. Graph or use Interval Table method to find the intervals x which satisfy the equation

 

Unit 4: Trigonometry

  • Radian Measure: Angle X is defined as the length, a, the arc that extends the angle divided by the radius, r, of the circle: X = a/r.
  • 2 Pi rad = 360 degrees or Pi rad = 180 degrees.
  • To convert degrees to radians, multiply degree measure with Pi/180.
  • To convert radian measure to degrees, multiply radian measure with 180/Pi.

 

  • You can use a calculator to calculate trigonometric ratios and special angles for an angle expressed in radian measure by setting the angle mode to radians.
  • You can determine the reciprocal trigonometric ratios for an angle expressed in radian measure by first calculating the primary trigonometric ratios then using the reciprocal key on the calculator.
  • You can also use the unit circle and special triangles to determine exact values for the trigonometric ratios of the special angles 0, Pi/6, Pi/3. Pi/4, and Pi/2.
  • You can use the unit circle along with the CAST rule to determine exact values for the trigonometric ratios of multiples of the special angles.

 

  • For Equivalent Trigonometric Expressions, you can use a right triangle to derive equivalent trigonometric expressions that form the cofunction identities, such as sinx = cos(Pi/2 – x).
  • You can use the unit circle along with transformations to derive equivalent trigonometric expressions that form other trigonometric identities, such as cos(Pi/2 + X) = -sinx
  • Given a trigonometric expression of a known angle, you can use the equivalent trigonometric expressions to evaluate trigonometric expressions of other angles.
  • You can use graphing technology to demonstrate that two trigonometric expressions are equivalent. Some of which are known as the Co-Functions Identities.

 

  • Compound Angle Formulas
    • You can develop compound angles formulas using algebra and the unit circle.
    • Once you have developed one compound angle formula, you can develop others by applying equivalent trigonometric expressions.
    • The compound angle, or addition and subtraction, formulas for sine and cosine are:
      • sin(X+Y) = sinXcosY+cosXsinY
      • sin(X-Y) = sinXcosY-cosXsinY
      • cos(X+Y) = cosXcosY-sinXsinY
      • cos(X-Y) = cosXcosY+sinXsinY
      • tan(X+Y) = tanX+tanY / 1-tanXtanY
      • tan(X-Y) = tanX-tanY / 1+tanXtanY
    • These identities can also be made into more identites:
      • cos2X = cos^2X – sin^2X
        • = 1 – 2sin^2X
        • = 2sin^2X – 1
      • sin2X = 2sinXcosX
      • tan2X = 2tanX / 1-tan^2X

 

  • Proving Trigonometric Identities:
    • A Trigonometric identity is an equation which trigonometric expressions that is true for all angles in the domain of the expressions on both sides.
    • One way to show that an equation is not an identity is to determine a counter example
    • To prove that an equation is an identity, treat each side of the equation independently and transform the expression on one side into the exact form of the expression on the other side.
    • The basic trigonometric identities are the Pythagorean identity, the quotient identity, the reciprocal identities, the compound angle formulas. You can use these identities to prove more complex identities.
    • Trigonometric identities can be used to simplify solutions to problems that result in trigonometric expressions.

 

Unit 5: Trigonometric Functions

  • Graphs of Sine, Cosine, and Tangent Functions
    • The graphs of y = sin x, y = cos x, and y = tan x are periodic.
    • The graphs of y = sin x and y = cos x are similar in shape and have an amplitude of 1 and a period of 2π
    • The graph of y = sin x can be transformed into graphs modeled by equations of the form y = sin x + c, y = sin (x – d), and y = sin kx. Similarly, the graph of y = cos x can be transformed into graphs modeled by equations of the form y = cos x + c, y = acos x, y = cos (x – d), and y = cos kx.
    • The graph of y = tan x has no amplitude because it has no maximum or minimum values. It is undefined at values of x that are odd multiples of π/2, such as π/2 and 3π/2.
    • The graph becomes asymptotic as the angle approaches these values from left and the right. The period of the function is π.

 

  • Graphs of Reciprocal Sine, Cosine, and Tangent Functions
    • The graphs of y = csc x, y = sec x, and y = cot x are periodic. They are related to the primary trigonometric functions as reciprocal graphs.
    • Reciprocal trigonometric functions are different from inverse trigonometric functions.
      • csc x means 1 / sin x, while sin-1 x asks you to find an angle that has a sine ratio equal to x.
      • sec x means 1 / cos x, while cos-1 x asks you to find an angle that has a cosine ratio equal to x.
      • cot x means 1 / tan x, while tan-1 x asks you to find an angle that has a tangent ratio equal to x.

 

  • Sinusoidal functions of the form f(x) = a sin[k(x - d)] + c and f(x) = a cos[k(x - d)] + c
    • The transformation of a sine or cosine function f(x) to g(x) has the general form g(x) = a f [k(x - d)] + c, where |a| is the amplitude, d is the phase shift, and c is the vertical translation.
    • The period of the transformed function is given by 2π / k. 
    • The k value of the function is given by 2π / period.

 

  • Solving Trigonometric Functions
    • Trigonometric equations can be solved algebraically by hand or graphically using technology.
    • There are often multiple solutions. Ensure that you find all solutions that lie in the domain of interest.
    • Quadratic trigonometric equations can often be solved by factoring.
    • Often, a trigonometric equation might need to be manipulated using trigonometric identities in order of it to be solved. Refer to notes on trigonometric identities here.

 

Unit 6: Logarithmic Functions

  • Exponential Functions
  • an exponential function of the form y = bx , b > 0, b not equal 1, has
    • a repeating pattern of finite differences
    • a rate of change that is increasing proportional to the function for b > 1
    • a rate of change that is decreasing proportional to the function for 0 < b < 1
  • An exponential function of the form  y = bx , b > 0, b not equal 1,
    • has a domain X E R
    • a range Y E R, Y > 0
    • a y-intercept of 1
    • has a horizontal asymptote at y = 0
    • is increasing on its domain when b > 1
    • is decreasing on its domain when 0 < b < 1
  • The inverse of  y = bx is a function that can be written as x = by.
    • has a domain of X E R, x > 0
    • a range of Y E R
    • a x-intercept of 1
    • has vertical asymptote at x = 0
    • is a reflection of  y = bx about the line y = x
    • is increasing on its domain when b > 1
    • is decreasing on its domain when 0 < b < 1

 

  • Logarithms
  • a logarithmic function is the inverse of the exponential function
  • The value of logbx is equal to the exponent to which the base, b, is raised to produce product x
  • Exponnetial equations can be written in logarithmic form, and vice versa
    • y = b^x   <->  x = logby
    • y = logbX   <->   x = b^y
  • Exponential and logarithmic functions are defined only for positive values of the base that are not equal to one. In other words, b not = 1, and x > 0.
  • The logarithm of x to base 1 is only valid when x = 1, in which base y has an infinite number of solutions and is not a function.
  • Common logarithms are logarithms wit ha base of 10. It is not necessary to write the base for common logarithms: logx means log base 10 x.

 

  • Transformations of Logarithmic Functions
  • The techniques for applying transformations to logarithmic functions are the same for those used for other functions:
    • y = log x + c
      • translate up c units if c > 0
      • translate down c units if c < 0
    • y = log( x – d)
      • translate right d units if d > 0
      • translate left d units if d < 0
    • y = a log x
      • stretch vertically by a factor of |a| if |a| > 1
      • Compress vertically by a factor of |a| if |a| < 1
      • Reflect in the x-axis if a < 0
    • y = log (kx)
      • compress horizontally by a factor of |1/k| if |k| < 1, k not = 0.
      • Reflect in the y axis if k < 0.
  • When all transformations are combined, they follow the form:

 

f(x) = a log[k(x-d)] + c

 

  • Power of logarithms
  • The power of logarithms states that logbxn = n logbx for b > 0, b not = 1, x > 0, and n ER
  • Any logarithm can be expressed in terms of common logarithms using the change of base formula:
  • log base b m = log m / log b, b > 0, b not = 1, m > 0

 

  • Manipulating Exponential Functions
  • Exponential functions and expressions can be expressed in different ways by changing the base
  • Changing the base of one or more exponential expressions is a useful technique for solving exponential equations

 

 

  • Solving Exponential Functions
  • An equation maintains balance when the common logarithm is applied to both sides
  • The power of logarithms is a useful tool for solving a variable that appears as a part of an exponent
  • When a quadratic equation is obtained, methods such as factoring and applying the quadratic formula may be useful.
  • Some algebraic methods of solving exponential functions lead to extraneous roots, which are not valid solutions to the original equation

 

 

  • Laws of logarithms
  • The product law of logarithms states that logbX + logb Y = logb(XY) for b > 0, b not = 1, x > 0, y > 0
  • The quotient law of logarithms states that logbX – logbY = logb(X/Y) for b > 0, b not = 1, x > 0, y > 0

 

 

  • Applications of Exponential/ Logarithmic Functions
  • Interest rates:

A = (i + 1)t

Where A is the amount, i is the interest, and t is the time.

 

  • Population

P = A1 (i)t

Where P is the population, A1is the initial amount, i is the amount increase, and t is time.

 

  • Half Life:

Ao = Ai (1/2)t/h

Where Ao is the final amount, Ai is the initial amount, t is time, and h is the time interval

 

  • pH Levels:

pH = -log (H+)

Where H+ is the concentration of hydrogen ions

Every Integer increment in pH is 10 times more acidic

 

  • Sound Intensity:

L = 10log(I / Io)

Where L is the loudness, I is the intensity, and Io is the sound that is barely audible

Every Integer increment in L, decibels, is 10 times more intense

 

  • Earthquakes:

M = log(E/ Io)

Where M is the Richter Number, E is the earthquake’s intensity, and Io is the intensity of a referenced earthquake.

Every Integer increment in M, Richter readings, is 10 times greater the earthquake intensity.

 

Unit 7: Combining Functions

  • Adding and Subtracting Functions
    • Functions that are added/subtracted are called combined functions.
    • Points on the functions can be added by adding y-coordinates at each point along the x axis of both functions.
    • Same goes for subtracting functions at each point.
    • Domain of the sum or difference is the domain of the functions being added or subtracted.

 

  • Multiplying and Dividing Functions
    • Function of the form y = f(x) g(x) is the product of f(x) and g(x)
    • Function of the form y = f(x) / g(x) is the quotient of the function f(x) and g(x) where g(x) cannot equal 0.
    • Domain of these combined functions is the domain of the functions being combined.
    • When dividing functions, the denominator has a restriction where it cannot equal 0.

 

  • Composite Functions
    • You can combine functions using the notation: f(g(x)) where f(x) depends on the answer of g(x). Also known as (f o g)(x).
    • To do this, you must find the value for g(x) then substitute that into f(x) to get your answer.

 

  • Inequalities of combined functions
    • Similarly to finding inequalities for normal functions, simply move everything to one side, factor then solve. Or graph and see where it meets the criteria.