Ch. 9.8 Cubic Functions & Ch. 9.8 Rational Expressions Learning Intentions: Explore general patterns & characteristics of cubic functions.

Learn formulas that model the areas of squares & the volumes of cubes.

Explore the graphs of cubic functions & transformations of these graphs.

Write the equation of a cubic function from its graph.

Tuesday, 3/28 : Ch. 9.8 Cubic Functions

~ Ch. 9 Packet p.67 #(1-6)

Thursday, 3/30 : Ch. 9.8 Rational Expressions

~ Ch. 9 Packet p.68 #(1-3)

Cube Root Function:

𝒚 = 𝒙𝟑

Parent Functions &

Their Relationships

Cubic Function:

𝒚 = 𝒙𝟑

cubing function: the function 𝒇 𝒙 = 𝒙𝟑, which gives the cube of a number. cube root: the cube root of a number 𝑎 is the number 𝑏 such that 𝑎 = 𝑏3. the cube root of 𝑎 is denoted 𝒂𝟑 . ex.) The cube root of 64 equals 4. 64

3= 𝟒

The cube root of 16 equals ≈ 2.52. 𝟏𝟔3

≈ 2.5198421 perfect cube: a number, 𝒂, that is equal to the cube of an integer. Given: 𝒂 = 𝒃𝟑= 𝑏 · 𝑏 · 𝑏 ex.) -125 is a perfect cube because -125 = (−5)3

Perfect cubes: 𝒂: { 0, ±1, ±8, ±27, ±64, ±125, ±216, ±343, ±512, ±729, ±1000…}

Cube roots: b: { 0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10…} 𝒏 𝒏𝟐2

𝒏𝟑3

𝒙𝟑

(-2)3

(−1)3

(0)3

(1)3

(2)3

(3)3

(h, k)

What should we call the point (h, k) of a

cubic function? If it’s not a vertex (pt. of parabola’s

directional ∆), then what is it??

A square is to a rectangle

like a cube is to a _______ .

Transformation of Cubic Parent Function 𝒚 = 𝒙𝟑

Replace x → 𝑥 − ℎ

𝑎 & y →

𝑦 −𝑘

𝑏

𝒚 = 𝒙𝟑

𝑦 −𝑘

𝑏= (

𝑥 − ℎ

𝑎)𝟑

𝐲 = 𝒃(𝒙 − 𝒉

𝒂)𝟑 + 𝒌

h = horizontal shift k = vertical shift a = horizontal dilation b = vertical dilation

(h, k) = the inflection point — or, the center of rotational symmetry

A square is to a rectangle

like a cube is to a _______ .

Answer: PRISM.

Prism: a solid geometric figure whose two end faces are similar, equal, & parallel, however, the sides are parallelograms, but NOT equal to the ends.

l ≠ 𝒘 ≠ 𝒉

𝑽 = 𝒍𝒘𝒉

l = 𝒘 = 𝒉

𝑽 = 𝒔𝟑

Transformation of Cubic Parent Function 𝒚 = 𝒙𝟑

Replace x → 𝑥 − ℎ

𝑎 & y →

𝑦 −𝑘

𝑏

𝒚 = 𝒙𝟑

𝑦 −𝑘

𝑏= (

𝑥 − ℎ

𝑎)𝟑

𝐲 = 𝒃(𝒙 − 𝒉

𝒂)𝟑 + 𝒌

h = horizontal shift k = vertical shift a = horizontal dilation b = vertical dilation

Cubic Function Transformations Parent Cubic Function

local maximum

(5, 2,000)

End behavior:

As x increases,

y also increases.

local minimum

(15, 0)

End behavior:

As x decreases,

y also decreases.

General Cubic Function → Factored Form ??? 𝒂𝑥𝟑 + 𝑏𝑥2 + 𝑐𝑥 + 𝒅 → 𝒂(𝑥 − 𝒓𝟏)(𝑥 − 𝒓𝟐)(𝑥 − 𝒓𝟑)

RECALL: General Quadratic Function → Factored Form 𝒂𝑥𝟐 + 𝑏𝑥 + 𝑐 → 𝒂(𝑥 − 𝒓𝟏)(𝑥 − 𝒓𝟐)

local maximum

(5, 2,000)

End behavior:

As x increases,

y also increases.

local minimum

(15, 0) AND a DOUBLE ROOT

End behavior:

As x decreases,

y also decreases.

General Cubic Function → Factored Form 𝒂𝑥𝟑 + 𝑏𝑥2 + 𝑐𝑥 + 𝒅 → 𝒂(𝑥 − 𝒓𝟏)(𝑥 − 𝒓𝟐)(𝑥 − 𝒓𝟑) 𝟒𝑥𝟑 − 120𝑥2 + 900𝑥 + 𝟎 → 𝟒(𝑥 − 𝒓𝟏)(𝑥 − 𝒓𝟐)(𝑥 − 𝒓𝟑) factor out GCF & use M:A find roots & use factored form equation 4x(𝑥𝟐 − 30𝑥 + 225) = 𝟒(𝑥 − 𝟎)(𝑥 − 𝟏𝟓)(𝑥 − 𝒓𝟑) 4x(𝑥 − 15)(𝑥 − 15) = 𝟒(𝑥 − 𝟎)(𝑥 − 𝟏𝟓)𝟐

Thus, 𝒚 = 𝟒𝒙𝟑 − 𝟏𝟐𝟎𝒙𝟐 + 𝟗𝟎𝟎𝒙

also equals 𝒚 = 𝟒𝒙(𝒙 − 𝟏𝟓)𝟐

in factored form.

When a-value is

positive

a.) Identify the zeros or roots of the cubic function. 𝒓𝟏= ___ 𝒓𝟐= ___ 𝒓𝟑= ___

b.) Find the y-intercept. Use this point to solve for the dilation, 𝒂.

c.) Write the factored form of the function. y = 𝒂(𝑥 − 𝒓𝟏)(𝑥 − 𝒓𝟐)(𝑥 − 𝒓𝟑)

Writing the Factored Form of a Cubic Function (given a graph)

SOLUTIONS: Zeros or Roots: a.) 𝑟1 = -5, 𝑟2 = 3 & 𝑟3 = 7 -> y = 𝒂(𝑥 − −𝟓)(𝑥 − 𝟑)(𝑥 − 𝟕) y-intercept: b.) (0, d) = (0, 105)

Factored form cubic function: c.) y = 1(x + 5)(x – 3)(x – 7)

Let (x,y) = (0, 105) ; solve for a-value.

y = 𝒂(𝑥 + 𝟓)(𝑥 − 𝟑)(𝑥 − 𝟕)

105 = 𝒂(0 + 5)(0 – 3)(0 – 7)

1 = 𝒂

Thus,

The factored form cubic function is:

y = 1(x + 5)(x – 3)(x – 7)

SOLUTIONS: Zeros or Roots: a.) 𝑟1 = 1, 𝑟2 = -3 & 𝑟3 = -3 -> y = 𝒂(𝑥 − 𝟏)(𝑥 − −𝟑)(𝑥 − −𝟑) y-intercept: b.) (0, d) = (0, -18)

Factored form cubic function: c.) y = 2(x – 1)(x + 3)𝟐

Let (x,y) = (0, -18) ; solve for a-value.

y = 𝒂(𝑥 − 𝟏)(𝑥 + 𝟑)(𝑥 + 𝟑)

-18 = 𝒂(0 – 1)(0 + 3)𝟐

-18 = -9𝒂

2 = 𝒂

Thus,

The factored form cubic function is:

y = 2(x – 1)(x + 3)𝟐

Ch. 9.8 Rational Expressions – Part II p.544 #13.) Simplify each rational expression completely. State any restrictions on the variable.

a.) 𝒙 + 𝟒

𝒙 + 𝟐 ·

𝒙𝟐+ 𝟒𝒙 + 𝟒

𝒙𝟐 − 𝟏𝟔

b.) 𝒙𝟐+ 𝟐𝒙

𝒙𝟐 − 𝟒 ÷

𝒙𝟐

𝒙𝟐 −𝟔𝒙 + 𝟖

c.) 𝒙

𝒙𝟐+ 𝟔𝒙 + 𝟗 +

𝟏

𝒙 + 𝟑

d.) 𝒙 − 𝟏

𝒙𝟐 − 𝟏 −

𝟒

𝒙 +𝟏

SOLUTIONS: p.544 #13.) Simplify each rational expression completely. State any restrictions on the variable.

a.) 𝒙 + 𝟒

𝒙 + 𝟐 ·

𝒙𝟐+ 𝟒𝒙 + 𝟒

𝒙𝟐 − 𝟏𝟔

𝑥 + 4

𝑥 + 2 ·

(𝑥+2)(𝑥+2)

(𝑥+4)(𝑥−4)

(𝒙+𝟐)

(𝒙−𝟒)

b.) 𝒙𝟐+ 𝟐𝒙

𝒙𝟐 − 𝟒 ÷

𝒙𝟐

𝒙𝟐 −𝟔𝒙 + 𝟖

𝑥(𝑥+ 2)

(𝑥+4)(𝑥−4) ×

(𝑥−4)(𝑥−2)

𝒙𝟐

(𝑥+ 2)(𝑥−2)

𝑥(𝑥+4) or

𝒙𝟐− 𝟒

𝒙𝟐+ 𝟒𝒙

c.) 𝒙

𝒙𝟐+ 𝟔𝒙 + 𝟗 +

𝟏

𝒙 + 𝟑

𝑥

(𝑥+3)(𝑥+3) +

1

𝑥 + 3

𝑥

(𝑥+3)(𝑥+3) +

1

𝑥 + 3∙ (

𝑥+3

𝑥+3)

𝑥+1(𝑥+3)

(𝑥+3)2

𝑥+𝑥+3

(𝑥+3)2

𝟐𝒙+𝟑

(𝒙+𝟑)𝟐

d.) 𝒙 − 𝟏

𝒙𝟐 − 𝟏 −

𝟒

𝒙 +𝟏

𝑥 − 1

(𝑥−1)(𝑥+1) −

4

𝑥 +1

𝑥 − 1

(𝑥−1)(𝑥+1) −

4

𝑥 +1∙ (

𝑥−1

𝑥−1)

𝑥 − 1

(𝑥−1)(𝑥+1) −

4(𝑥−1)

𝑥2+1

𝟏(𝑥 − 1)−4(𝑥−1)

(𝑥−1)(𝑥+1)

−𝟑

𝒙 + 𝟏

D:{all Real x’s s.t.

x≠ -2, -4 or 4}

D:{all Real x’s s.t.

x≠ 0, or ±4}

D:{all Real x’s s.t.

x≠ -3} D:{all Real x’s s.t.

x≠ -1 or 1}

Lesson 9.8 Rational Expressions State any restrictions on the variable. Reduce each rational expression to lowest terms. p.68 #1e.) #2i.)

𝒙𝟐− 𝟓𝒙 − 𝟔

𝒙𝟐+ 𝟒𝒙 + 𝟑 ÷

𝒙𝟐− 𝟒𝒙 − 𝟏𝟐

𝒙𝟐+ 𝟓𝒙 + 𝟔 𝟒 + 𝟐𝟎𝒙

𝟐𝟎𝒙

SOLUTIONS: Lesson 9.8 Rational Expressions State any restrictions on the variable. Reduce each rational expression to lowest terms.

p.68 #1e.) #2i.) 𝒙𝟐− 𝟓𝒙 − 𝟔

𝒙𝟐+ 𝟒𝒙 + 𝟑 ÷

𝒙𝟐− 𝟒𝒙 − 𝟏𝟐

𝒙𝟐+ 𝟓𝒙 + 𝟔 𝟒 + 𝟐𝟎𝒙

𝟐𝟎𝒙

Factor out GCF from numerator. Factor that same value from the denominator. Reduce. Warning: DO NOT EVER divide only part of the numerator by the denominator.

i.e. 𝟏+5

5≠

1+5

5≠ 1 X

(1+5)

5=

6

5= 𝟏. 𝟐 √

𝟒(𝟏 + 𝟓𝒙)

𝟒(𝟓𝒙)

𝟒(𝟏+𝟓𝒙)

𝟒(𝟓𝒙)

𝟏 + 𝟓𝒙

𝟓𝒙

𝑥2− 5𝑥 − 6

𝑥2+ 4𝑥 + 3 ×

𝑥2+ 5𝑥 + 6

𝑥2− 4𝑥 − 12

(𝑥+1)(𝑥−6)

(𝑥+1)(𝑥+3) ×

(𝑥+3)(𝒙+𝟐)

(𝑥−6)(𝒙+𝟐)

(𝑥 + 1)(𝑥 + 3)

(𝑥 + 1)(𝑥 + 3)

(𝑥 − 6)(𝒙 + 𝟐)

(𝑥 − 6)(𝒙 + 𝟐)

1

Multiply by the

Reciprocal.

Factor.

Reduce.