1. simplify and evaluate each of the following, where …...mhf 4u lesson 4.0 simplifying rational...
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MHF 4U Lesson 4.0 Simplifying Rational Expressions
1. Simplify and evaluate each of the following, where x = 4. State any restrictions.
a) 36
652
2
x
xx b)
4
28112
x
xx
2. Simplify each expression and state any restrictions.
a) 105
3
3
442
a
a
a
aa b)
25152
372
25
3522
2
2
2
xx
xx
x
xx
3. Simplify each expression and state any restrictions on the variables.
a) 4
7
2
3
aa
a
b) 96
3
65
422
xxxx
Pg. 246 # 2 – 5, 7
MHF 4U Investigation 4.1 Graphs of Reciprocal Functions
INVESTIGATION 1
1. Sketch f(x) = x + 2 and g(x) = 2
1
x.
2. Complete the table below.
CHARACTERISTIC f(x) = x + 2 g(x) = 2
1
x.
x-intercepts and/or
asymptotes
interval(s) on which the
graph is above the x-axis
interval(s) on which the
graph is below the x-axis
interval(s) on which the
function is increasing
interval(s) on which the
function is decreasing
x-value(s) where
the y-value is 1
x-value(s) where
the y-value is -1
1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x
1
2
3
4
5
6
7
8
9
–1
–2
–3
–4
–5
–6
–7
–8
–9
y
INVESTIGATION 2
3. Sketch f(x) = 42 x and
g(x) = 4
12 x
.
4. Complete the table below.
Use a TI-83 to see how you did.
Use the window as shown below.
CHARACTERISTIC f(x) = 42 x g(x) = 4
12 x
.
x-intercepts and/or
asymptotes
interval(s) on which the
graph is above the x-axis
interval(s) on which the
graph is below the x-axis
interval(s) on which the
function is increasing
interval(s) on which the
function is decreasing
x-value(s) where
the y-value is 1
x-value(s) where
the y-value is -1
1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x
1
2
3
4
5
6
7
8
9
–1
–2
–3
–4
–5
–6
–7
–8
–9
y
Pg. 254 # 1, 2, 3, 5doso, 6, 7, 11, 12
SUMMARY
MHF 4U Investigation 4.2 – Part I Rational Functions
PART I - Use desmos to graph the following functions using the given window, then draw a sketch.
Row A Row B Row C
21
1
x
x
21
2
x
x
1x
x
check the point at x = -1
1
1 2
x
x
x
x
2
1 2
x
x 1
check the point at x = -1
1 2 3 4 5 6–1–2–3–4–5–6 x
2
4
6
8
10
12
–2
–4
–6
–8
–10
–12
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
0.5
1
1.5
2
2.5
–0.5
–1
–1.5
–2
–2.5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
PART II - Describe the characteristics of the graphs you created in PART I.
21
1
x
x
1
1 2
x
x 21
2
x
x
x
x
2
1 2
1x
x
x
x 1
zeros
asymptotes
domain
range
Continuous
Everywhere?
Undefined @?;
Reason
End
Behaviours
y
xAs
y
xAs
,
,
y
xAs
y
xAs
,
,
y
xAs
y
xAs
,
,
y
xAs
y
xAs
,
,
y
xAs
y
xAs
,
,
y
xAs
y
xAs
,
,
Function?
Reason
PART III - PART F from investigation on Page 259
(i) (ii) (iii)
check the point at x = 1
check the point at x = -1
(iv) (v) (vi)
(vii) (viii)
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
PART IV - PARTS G, H, I, N from pgs. 259 - 260
Question
G
Functions with
Holes
Functions with
V.A.
H
What do you notice about 1
15.0)(
2
x
xxf ?
Any other functions with an Oblique Asymptote?
I
Connection between degrees of numerator and denominator?
Horizontal Asymptote:.
Oblique Asymptote:
N
(i)
HOLE:
(ii)
VERTICAL
ASYMPTOTE:
(iii)
HORIZONTAL
ASYMPTOTE:
(iv)
OBLIQUE (SLANT)
ASYMPTOTE:
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
HW Day 1: Finish Parts I – IV of this investigation
MHF 4U Investigation 4.2 – Part II Rational Functions
Day 2: Pg. 262 # 1, 2a-d, 3
MHF 4U Lesson 4.3 Rational Functions in the Form f (x)
ax b
cx d
Property Example: f (x)
4 x 3
2x 1
x-intercepts
y-intercepts
Vertical
Asymptotes
Horizontal
Asymptotes
The two branches of the graph of a function in the form f (x)
ax b
cx d are equidistant from the point
of intersection of the vertical and horizontal asymptotes.
Ex. Sketch the graph of f (x)
4 x 3
2x 1.
1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x
1
2
3
4
5
6
7
8
9
–1
–2
–3
–4
–5
–6
–7
–8
–9
y
Pg. 272 # 1 – 5, 8, 9, 10
MHF 4U Lesson 4.4 Solving Rational Equations
ALGORITHM
Factor everything that is factorable.
Express using the Lowest Common Denominator (LCD).
Set = 0 or cross multiply (eliminates step )
Eliminate the denominator by multiplying both sides by the LCD.
Solve for the variable. Check restrictions and eliminate any extraneous solutions.
Check your answer(s).
Ex. 1 Solve each of the following. x R
a)
x 2
x
x 4
x 6 b)
x
x 2
1
x 4
2
x2 6x 8
c)
x x2 5
x2 1
x2 x 2
x 1
1 2 3 4 5 6 7–1–2–3–4–5–6–7 x
1
2
3
4
5
6
7
–1
–2
–3
–4
–5
–6
–7
y
Ex. 2 The chocolate company that makes Black Magic ® chocolates has two packing machines.
Machine A takes s minutes to fill a case and Machine B takes s + 10 minutes. Working
together, the two machines take 15 minutes to fill a case. About how long does each
machine take to fill a case?
Ex. Find the equation of the oblique asymptote and determine how the curve approaches
the OA as x
2
13)(
2
x
xxxf
Pg. 285 # 1, 3, 4, (5 – 7)sodo, 11, 12, 13
MHF 4U Lesson 4.5 SOLVING RATIONAL INEQUALITIES
Set < , > , or 0
Factor everything that is factorable.
Express using the Lowest Common Denominator (LCD).
Find any zeros using the numerator.
Find any vertical asymptotes using the denominator.
Construct an interval table based on the zeros and VA's and holes, if they exist
Determine where f(x) < 0 , f(x) > 0 , f(x) 0 or f(x) 0 depending on equation from .
Check your answer and state the solution.
Ex. 1 Solve. x R
a)
x2 3x 2
x2 16 0 b)
2x2 4x 30
(x2 5)(x 2)2 0
c)
2x 1
x 7
x 1
x 5
Pg. 295 # 1, 2, (4 – 6)sodo, 9, 11