2.1, 2.2: Radical Functions and the Square Root of a Functiondobsonmath.weebly.com/uploads/1/1/8/0/11809374/pc_12_ch_2__3_n… · Math 12 Pre-Calculus Chapter 2 & 3: ... Radical Functions

Math 12 Pre-Calculus Chapter 2 & 3: Radical & Polynomial Functions

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2.1, 2.2: Radical Functions and the Square Root of a Function

We have already studied the base function

y x

Some general characteristics of the base radical function: Domain: _______________________ Range: _______________________ Start point: _____________________ Endpoint: ____________________ Shape: ______________________________________________________________ We can graph any radical function using transformations from Ch. 1:

y a b(x h) k

a = vertical stretch

1/b = horizontal stretch

h = horizontal translation (opposite of how it appears)

k = vertical translation

Ex. 1: Sketch the graph of

y 3 2x using transformations

Ex. 2: Determine 2 possible equations to represent the function shown:


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We can determine how two functions

y f (x) and

y f (x) , are related by comparing their outputs.

For

y 2x1, multiply x by 2 and add 1.

For

y 2x 1 , multiply x by 2, add 1, and take the square root.

In this case,

y 2x 1 is the square root of the function of

y 2x1.

Ex. 3: Graph the two functions above. State their domain and range, as well as any invariant points.

In General for

y f (x) and

y f (x) :

y f (x)

Domain Values of x for which

f (x) 0

Range Corresponding square roots for defined domain.

Invariant Point(s) Where

f (x) 0 or

f (x) 1

A square root function will only exist where the output (y) values are positive.

Ex. 4: Identify and compare the domains and ranges of the functions

y x2 5 and

y x25 .

x [ , ] y[ , ]

Y1 = Y2 =


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Ex. 5: Using the graph of

y f (x), sketch the graph of

y f (x) .

2.3: Solving Radical Equations Graphically Last year, we learnt how to solve radical equations algebraically, by isolating the radical term and squaring both sides. Now, we will also be able to solve a radical equation by graphing a related function and finding its zeros (x-intercepts).

Ex. 1: a) Determine the root(s) of

x530 algebraically.

b) Graph the related function

y x 53 to determine the x-intercept(s).


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Ex. 2: Solve the equation

x5 x3 both algebraically and graphically.

Ex. 3: a) Solve the equation

3x2 5 x 4 using technology. Express your answer to the nearest tenth. b) Verify your solution algebraically.

x [ , ] y[ , ]

Y1 =


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3.1 (part 1): Characteristics of Polynomial Functions Complete questions 1 – 8 on pages 106 – 107 of your textbook. Record your answers below.

For Questions # 1 – 3, complete the following tables. Note: End Behaviour is the quadrants (I, II, III, or IV) that the function starts and ends in, read from left to right. The first set is completed for you.

Set Function End Behaviour Degree Constant

Term Leading

Coefficient

Number of x-

Intercepts A Linear

III, I

1

0

1

1

Quadratic

Cubic

Quartic

Quintic

B Linear

Quadratic

Cubic

Quartic

Quintic

C Linear

Quadratic

Cubic

Quartic

Quintic


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D

Quadratic

Cubic

Quartic

Quintic

Similar Graphs How are they similar?

y x

y x

y x2

y x2

5. a) How are the graphs and equations of linear, cubic and quintic functions similar?

b) How are the graphs and equations of quadratic and quartic functions similar? c) Describe the relationship between the end behaviours of the graphs and the degree of the corresponding function.

6. What is the relationship between the sign of the leading coefficient of a function equation and the end behaviour of the graph of the function?

7. What is the relationship between the constant term in a function equation and the position of the graph of the function?

8. What is the relationship between the minimum and maximum number of x-intercepts of the graph of a function with the degree of the function?


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3.1 (part 2): Characteristics of Polynomial Functions

A polynomial function is a function of the form:

f (x) anxn an1xn1 ...a1xa0

where n is a whole number, and a is a real number. in other words, the exponents cannot be a fraction, or negative. Recall that the degree of a function is the highest power of x. The leading coefficient (a) is the coefficient of the highest power of x. Recall the quadrant numbering system:

Degree 0: Constant Function Degree 1: Linear Function Degree 2: Quadratic Function

Even degree Max. of x-intercepts =

Degree 3: Cubic Function Degree 4: Quartic Function Degree 5: Quintic Function

Notice the END BEHAVIOUR of each even degree function and odd degree function is the same! EVEN Degree: if a > 0: II/I ODD Degree: if a > 0: III/I if a < 0: III/IV if a < 0: II/IV


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Ex. 1: Identify the following characteristics for each polynomial function: • the type of function and whether it is of even or odd degree • the end behaviour of the graph of the function • the number of possible x-intercepts • whether the function will have a maximum or minimum value • the y-intercept

a) g (x) = -x4 + 2x2 + 7x - 5 b) f (x) = 2x5 + 7x3 + 12

Ex, 2: Given the polynomial y = -2(x + 1)2(x - 2)(x - 3)2, determine the following without graphing.

a) Describe the end behaviour of the graph of the function.

b) Determine the possible number of x-intercepts of the function. c) Determine the y-intercept of the function. d) Now, use graphing technology to create a sketch of the graph.

x [ , ] y[ , ]

Y1 =


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3.2, 3.3: Remainder and Factor Theorems Recall that we can divide two polynomials using long division or synthetic division. When we perform this division, we can write a division statement:

P(x) (xa)Q(x) R where: P(x) = dividend (polynomial being divided)

(x-a) = divisor (polynomial dividing) Q(x) = quotient (result of division) R = remainder REMAINDER THEOREM: when a polynomial P(x) is divided by a binomial of the form x-a, the remainder is P(a).

Ex. 1: Determine the remainder when

P(x) x3 10x6 is divided by

x4 using:

a) long or synthetic division. b) the remainder theorem. c) Write the division statement for the above division. FACTOR THEOREM: if P(a)=0 then x-a is a factor of the polynomial P(x).

Ex. 2: Which binomials are factors of the polynomial

P(x) x3 3x2 x3 ? Justify your answers.

a)

x1 b)

x1 c)

x3 d)

x3


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The integral zero theorem tells us to check the factors of the constant term to find a factor of a polynomial.

Ex. 3: Factor

2x35x2

4x3 completely. To factor any polynomial:

1. Find an initial factor by finding a value where P(a)=0. Check all factors of the constant term. 2. Divide the polynomial by the initial factor using synthetic or long division. 3. Factor the remaining quotient by the factor theorem (again), or using factoring methods for

quadratics.

Ex. 4: Factor

x45x3

2x220x24 completely.

Why will it be useful to factor these polynomials?


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3.4: Equations and Graphs of Polynomial Functions As we have seen before with quadratic functions, the zeros of a polynomial function correspond to the x-intercepts of the graph and to the roots of the corresponding equation f(x) = 0. Ex. 1: State the degree and the zeros of the polynomial function:

a)

f (x) (x1)(x2)(x4) b)

g(x) x(x1)2(x2)

If a polynomial has a factor x-a that is repeated n times, then x = a is a zero of multiplicity n. In Ex.1b), the zero at x = 1 has multiplicity 2. What does this look like on a graph? Multiplicity 1 Multiplicity 2 Multiplicity 3 Ex. 2: Sketch the graph of each polynomial function without graphing technology.

a)

f (x) x(x2)3(x4)


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b)

y 2x3 6x4

Ex. 3: Given the graph of a polynomial y=f(x), determine a possible equation. State the roots of the equation.

The graph of a function

y a(b(xh))n k is obtained by applying transformations to the general

polynomial function

y xn . The effects are the same as in Chapter 1.

k = h = a = b =


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Ex. 4: The graph of

y x3 is transformed to obtain the graph of

y 2(4(x1))3 3.

a) State the parameters and describe the corresponding transformations. b) Copy and complete the table to show what happens to the given points under each transformation.

y x3

y (4x)3

y 2(4x)3

y 2(4x1)3 3

(-2, -8)

(-1, -1)

(0, 0)

(1, 1)

(2, 8)

c) Sketch the graph of

y 2(4(x1))3 3.

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2.1, 2.2: Radical Functions and the Square Root of a Functiondobsonmath.weebly.com/uploads/1/1/8/0/11809374/pc_12_ch_2__3_n… · Math 12 Pre-Calculus Chapter 2 & 3: ... Radical Functions