1.1 lines increments if a particle moves from the point (x 1,y 1 ) to the point (x 2,y 2 ), the...
TRANSCRIPT
1.1 Lines
Increments
If a particle moves from the point (x1,y1) to the point (x2,y2), the increments in its coordinates are
1212 ΔandΔ yyyxxx
1.1 Lines
Slope
Let P1= (x1,y1) and P2= (x2,y2) be points on a nonverticalline L. The slope of L is
12
12
Δ
Δ
run
rise
xx
yy
x
ym
P1(x1,y1)
P2(x2,y2)
Q(x2,y1)
Δy
Δx
1.1 Lines
Theorem: If two lines are parallel, then they have the same slope and if they have the same slope, then they are parallel.
m1 m2
L1 L2
slope m1 slope m2
θ1 θ2
1 1
Proof: If L1 || L2, then θ1= θ2
and m1= m2. Conversely, ifm1 = m2, then θ1= θ2 and L1 || L2.
1.1 Lines
Theorem: If two non vertical lines L1and L2 are perpendicular, then their slopes satisfy m1m2 = -1 and conversely.
L1L
2
Slope m2Slope m1
A
C
B
Proof: Δ ADC ~ ΔCDB
θ1
θ1
θ2h
D a
m1 = tan θ1 = a/hm2 = tan θ2 = -h/a
so m1m2 =(a/h)(-h/a) = -1
1.1 Lines
Equations of lines• Point-Slope Formula y = m(x – x1) + y1
• Slope-Intercept form y = mx + b• Standard form Ax + By = C• y = a Horizontal line slope of zero• x =a Vertical line no slope
1.1 Lines
Regression Analysis1. Plot the data2. Find the regression equation y = mx + b3. Superimpose the graph on the data points.4. Use the regression equation to predict y-values.
1.1 Lines
Coordinate Proofs1. State given and prove.2. Draw a picture.3. Label coordinates, use (0,0) if possible.4. Fill in missing coordinates.5. Use algebra to prove
• parallel/perpendicular-slope• equidistant-distance formula• bisect-midpoint
1.1 Lines
Prove the midpoint of the hypotenuseof a right triangle is equidistantfrom the three vertices.
A(0,0) C(b,0)
B(0,a)
Given: ΔBAC is a right triangleProve: AM = BM = CM
M(b/2,a/2)
4
a
4
b0
2
a0
2
bAM
2222
4
a
4
ba
2
a0
2
bBM
2222
4
a
4
b
2
a0
2
bbCM
2222
Since AM = BM = CM, themidpoint of the hypotenuseof a right triangle isequidistant from the three vertices
1.2 Functions and Graphs
Function
A function from a set D to a set R is a rule that assigns a unique element R to each element D.
y = f(x) y is a function of x
1.2 Functions and Graphs
Domain All possible x values
Range All possible y values
1.2 Functions and Graphs
x ),( 0
ax ),( aa
a bbxa ),( ba
a bbxa ],[ ba
open
closed
a b
a b bxa
bxa
],( ba
),[ ba
half opened
half opened
•y = mx
•Domain (-∞ , ∞)•Range (-∞ , ∞)
1.2 Functions and Graphs
•y = x2
•Domain (-∞ , ∞)•Range [0, ∞)
1.2 Functions and Graphs
•y = x3
•Domain (-∞ , ∞)•Range (-∞ , ∞)
1.2 Functions and Graphs
•y = 1/x
•Domain x ≠ 0•Range y ≠ 0
1.2 Functions and Graphs
xy
•Domain [0, ∞)•Range [0, ∞)
1.2 Functions and Graphs
1.2 Functions and Graphs
Function Domain Range
y = x ),( ),(
y = x2 ),( )0,[
y = |x| ),( )0,[
29 xy [-3,3] [0,3]
2 xy ),- 2[ )0,[
1.2 Functions and Graphs
Definitions Even Function, Odd Function
A function y = f(x) is aneven function of x if f(-x) = f(x) odd function of x if f(-x) = -f(x)
for every x in the function’s domain.
Even Function – symmetrical about the y-axis.Odd Function - symmetrical about the origin.
1.2 Functions and Graphs
Odd Function symmetrical about the origin.
Even Function symmetrical about the y-axis.
(x,y)
(-x,-y)
(-x,y) (x,y)
1.2 Functions and Graphs
Transformations
h(x) = af(x) vertical stretch or shrink
h(x) = f(ax) horizontal stretch or shrink
h(x) = f(x) + k vertical shift
h(x) = f(x + h) horizontal shift
h(x) = -f(x) reflection in the x-axis
h(x) = f(-x) reflection in the y-axis
1.2 Functions and Graphs
Piece Functions
112
1)(
2
xx
xxxf
1.2 Functions and GraphsPiece Functions
11
12
2||
)( 2
xx
xx
xx
xf
1.2 Functions and Graphs
Composite Functions f(g(x))
f(x) = x2, g(x) = 3x - 1
Find:1. f(g(2))2. g(f(-1))3. g(f(x))4. f(g(x))
•25•2•3x2 – 1•(3x – 1)2 = 9x2 – 6x + 1
1.3 Exponential Functions
Definition Exponential Function
Let a be a positive real number other than 1,the function f(x) = ax is the exponential function with base a.
1.3 Exponential Functions
Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.
yxyx aa. a 1
yxy
x
aa
a. 2
xyyx aa. 3
xxx (ab)b. a 4
x
xx
b
a
b
a.
5
16 0 . a
x-x
a. a
17
q pq
p
a. a 8
Use the rules for exponents tosolve for x.
•4x = 128•(2)2x = 27
•2x = 7•x = 7/2
•2x = 1/32•2x = 2-5
•x = -5
1.3 Exponential Functions
•(x3y2/3)1/2
•x3/2y1/3
•27x = 9-x+1
•(33)x = (32)-x+1
•33x = 3-2x+2
•3x = -2x+ 2•5x = 2•x = 2/5
1.3 Exponential Functions
1.3 Exponential Functions
Domain: Range:Increasing for:Decreasing for:Point Shared On All Graphs:Asymptote:
(-∞, ∞)(0, ∞)
a > 10 < a < 1
(0, 1)y = 0
Properties of f (x) = ax
1.3 Exponential Functions
xexf )(
Natural Exponential Function where e is the natural base and e 2.718…
x
x xxe
11lim
1.3 Exponential Functions
Function f(x) = 2x h(x) = (0.5)x g(x) = ex
Domain
Range
Increasing or Decreasing
Point Shared On All Graphs
(-∞, ∞) (-∞, ∞) (-∞, ∞)
(0, ∞) (0, ∞) (0, ∞)
Inc. Dec. Inc.
(0, 1)
1.3 Exponential Functions
Use translation of functions to graph the following. Determine the domain and range of each.
1. f(x) = -5(x + 2) – 3
2. g(x) = (1/3)(x – 1) + 2
1.3 Exponential Functions
Definitions Exponential Growth, Exponential Decay
The function y = k ax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.
h
t
Obyy y new amountyo original amountb baset timeh half life
1.3 Exponential Functions
An isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.
(a) Find the amount remaining after t hours.
(b) Find the amount remaining after 60 hours.
(c) Estimate the amount remaining after 4 days.
(d) Use a graph to estimate the time required for the mass to be reduced to 0.1 g.
1.3 Exponential Functions
An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.
(a) Find the amount remaining after t hours.
(b) Find the amount remaining after 60 hours.
• a. y = yobt/h
• y = 2 (1/2)(t/15)
• b. y = yobt/h
• y = 2 (1/2)(60/15)
• y = 2(1/2)4
• y = .125 g
1.3 Exponential FunctionsAn isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.
(c.) Estimate the amount remaining after 4 days.
(d.) Use a graph to estimate the time required for the mass to be reduced to 0.01 g.
• c. y = yobt/h
• y = 2 (1/2)(96/15)
• y = 2(1/2)6.4
• y = .023 g
d.
1.3 Exponential Functions
A bacteria double every three days. There are 50 bacteria initially present
(a) Find the amount after 2 weeks.
(b) When will there be 3000 bacteria?
• a. y = yobt/h
• y = 50 (2)(14/3)
• y = 1269 bacteria
1.3 Exponential Functions
A bacteria double every three days. There are 50 bacteria initially present
When will there be 3000 bacteria?
• b. y = yobt/h
• 3000 = 50 (2)(t/3)
• 60 = 2t/3
•
Equations where x and y are functions of a third variable, such as t. That is,
x = f(t) and y = g(t).
The graph of parametric equations are called parametric curves and are defined by (x, y) = (f(t), g(t)).
1.4 Parametric Equations
1.4 Parametric Equations
Equations defined in terms of x and y. These may or may not be functions. Some examples include:
x2 + y2 = 4y = x2 + 3x + 2
1.4 Parametric Equations
ty
tx
3
21
Sketch the graph of the parametric equation for tin the interval [0,3]
t x y
0 1 0
1 -1 3
2 -3 6
3 -5 9
1.4 Parametric Equations
ty
tx
3
21
Eliminate the parameter t from the curve
xt 12
2
1 xt
2
13
xy
2
3
2
3 xy
t
Circle:If we let t = the angle, then:
cos sin 0 2x t y t t
Since: 2 2sin cos 1t t
2 2 1y x
2 2 1x y We could identify the parametric equations as a circle.
1.4 Parametric Equations
Ellipse: 3cos 4sinx t y t
cos sin3 4
x yt t
2 22 2cos sin
3 4
x yt t
2 2
13 4
x y
This is the equation of an ellipse.
1.4 Parametric Equations
The path of a particle in two-dimensional space can be modeled by the parametric equations x = 2 + cos t and y = 3 + sin t. Sketch a graph of the path of the particle for 0 t 2.
1.4 Parametric Equations
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50
How is t represented
on this graph?
1.4 Parametric Equations
t = 0
t =
1.4 Parametric Equations
Graphing calculators and other mathematical software can plot parametric equations much more efficiently then we can. Put your graphing calculator and plot the following equations. In what direction is t increasing?
(a) x = t2, y = t3 (b)
(c) x = sec θ, y = tan θ; -/2 < θ < /2
1;,ln ttytx
1.4 Parametric Equations
Parametric equations can easily be converted to Cartesian equations by solving one of the equations for t and substituting the result into the other equation.
1.4 Parametric Equations
(a) x = t2, y = t3
2
33
xxy,xt
2ytty 2lnln ytx
0;ln 22 xeyeyeyyx xxx
1.4 Parametric Equations
1,ln tfortytx(b)
(c) x = sec t, y = tan twhere -/2 < t < /2
Hint: sec2 θ – tan2 θ = 1
1.4 Parametric Equations
2222 tansec tyt,x
1tansec 2222 tt-yx
122 yx
1.4 Parametric Equations
Find a parametrization for the line segment with endpoints(2,1) and (-4,5).
x = 2 + at y = 1 + bt
when t = 1, a = -6when t = 1, b = 4
x = 2 – 6t and y = 1 + 4t
Cartesian Equationm = (5 – 1)/(-4 – 2) = -2/3
y = mx + b1 = (-2/3)(2) + bb = 7/3
y = (-2/3)x + 7/3
1.5 Functions and Logarithms
A function is one-to-one if two domain values do not have the same range value.
Algebraically, a function is one-to-one if f (x1) ≠ f (x2) for all x1 ≠ x2.
Graphically, a function is one-to-one if its graph passes the horizontal line test. That is, if any horizontal line drawn through the graph of a function crosses more than once, it is not one-to-one.
To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.
31
2y x 21
2y x 2x y
one-to-one not one-to-one not a function
(also not one-to-one)
1.5 Functions and Logarithms
1.5 Functions and Logarithms
Determine if the following functions are one-to-one.
(a) f (x) = 1 + 3x – 2x 4
(b) g(x) = cos x + 3x 2
(c)
(d)
2)(
xx eexh
xxf 5)(
1.5 Functions and Logarithms
The inverse of a one-to-one function is obtained by exchanging the domain and range of the function. The inverse of a one-to-one function f is denoted with f -1.
Domain of f = Range of f -1
Range of f = Domain of f -1
f −1(x) = y <=> f (y) = x
To prove functions areinverses show that
f(f-1(x)) = f-1(f(x)) = x
1.5 Functions and Logarithms
To obtain the formula for the inverse of a function, do the following:
1. Let f (x) = y.2. Exchange y and x.3. Solve for y.4. Let y = f −1(x).
Inverse functions:
11
2f x x
Given an x value, we can find a y value.
11
2y x
Switch x and y:
1 2 2f x x
Inverse functions are reflections about y = x.
Solve for y:
1.5 Functions and Logarithms
12
1 yx
yx2
11 22 xy
11
2f x x 1 2 2f x x
1.5 Functions and Logarithms
Prove f(x) and f-1(x) are inverses.
xxxxfxff 111)22(2
1)22())(( 1
xxxxfxff
2221
2
121
2
1))(( 11
1.5 Functions and Logarithms
You can obtain the graph of the inverse of a one-to-one function by reflecting the graph of the original function through the line y = x.
1.5 Functions and Logarithms
1.5 Functions and Logarithms
1.5 Functions and Logarithms
Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f.
Domain: (0, ∞)Range: (-∞, ∞)
1.5 Functions and Logarithms
Determine the formula for the inverse of the following one-to-one functions.
(a)
(b)
(c)
32)( 3 xxf2
13)(
x
xxh
xxg 3)(
1.5 Functions and Logarithms
The inverse of an exponential function is called a logarithmic function.
Definition: x = a y if and only if y = log a x
1.5 Functions and Logarithms
The function f (x) = log a x is called a logarithmic function.
Domain: (0, ∞)Range: (-∞, ∞)
Asymptote: x = 0 Increasing for a > 1
Decreasing for 0 < a < 1 Common Point: (1, 0)
Find the inverse of g(x) = 3x.
Definition: x = a y if and only if y = log a x
xxg 31 log)(
1.5 Functions and Logarithms
1. log a (ax) = x for all x 2. alog ax = x for all x > 03. log a (xy) = log a x + log a y4. log a (x/y) = log a x – log a y5. log a xn = n log a x
Common Logarithm: log 10 x = log xNatural Logarithm: log e x = ln x
All the above properties hold.
1.5 Functions and Logarithms
The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula.
a
xx
b
ba log
loglog
where b is any other appropriate base.
1.5 Functions and Logarithms
$1000 is invested at 5.25 % interest compounded annually.How long will it take to reach $2500?
1000 1.0525 2500t
1.0525 2.5t We use logs when we have an
unknown exponent.
ln 1.0525 ln 2.5t
ln 1.0525 ln 2.5t
ln 2.5
ln 1.0525t 17.9
17.9 years
In real life you would have to wait 18 years.
1.5 Functions and Logarithms
Example 7: Indonesian Oil Production (million barrels per year):
1960 20.56
1970 42.10
1990 70.10
Use the natural logarithm regression equation to estimate oil production in 1982 and 2000.
How do we know that a logarithmic equation is appropriate?
In real life, we would need more points or past experience.
1.5 Functions and Logarithms
1. Determine the exact value of log 8 2.2. Determine the exact value of ln e 2.3.3. Evaluate log 7.3 5 to four decimal places.4. Write as a single logarithm: ln x + 2ln y – 3ln z.5. Solve 2x + 5 = 3 for x.
1.5 Functions and Logarithms
1.6 Trigonometric Functions
The Radian measure of angle ACBat the center of the unit circle equalsthe length of the arc that ACB cutsfrom the unit circle.
sθ
rr
sθ
so
1 circle,unit for thebut
C
A
Bθ
sr
1.6 Trigonometric Functions
θ
terminal ray
initial ray
y
xy
x
rP(x,y)
x
r θ:
y
r θ:
y
x θ:
x
y θ:
r
xθ:
r
yθ:
secsecantcsccosecant
cotcotangenttantangent
coscosinesinsine
0
15
30
45
607590105
120
135
150
165
180
195
210
225
240255 270 285
300
315
330
345
1.6 Trigonometric Functions
(2,/4)
(5,5 /6)
(4, 11/6)
(-4, /2)
1.6 Trigonometric Functions
Let a point P have rectangular coordinates (x,y)and polar coordinates (r,). Then
sin
cos
ry
rx
0tan
222
xx
y
ryx
(1,0)
3
2,-
1
2
2
2,
2
2
1
2,
3
2
(-1,0)
(0,1)
(0,-1)
1
2,
3
2
1
2,
3
2
2
2,-
2
2
2
2,-
2
2
2
2,
2
2
3
2,1
2
3
2,1
2
3
2,-
1
2
1.6 Trigonometric Functions
60° 1
3
2
30°
45°
2
1
145°
A
CT
S
2
3,
2
1
0
1/2
2 /2
3 /2
1
1
3 /2
2 /2
1/2
0
0
3 /3
1
3
2
2
2 3 /3
1
1
2 3 /3
2
2
3
1
3 /3
0
3 /2
2 /2
1/2
0
1/2
2 /2
3 /2
1
3
1
3 /3
0
2 3 /3
2
2
2
2
2 3 /3
1
3 /3
1
3
1/2
2 /2
3 /2
1
3 /2
2 /2
1/2
0
3 /2
2 /2
1/2
0
1/2
2 /2
3 /2
1
3 /3
1
3
3
1
3 /3
0
2
2
2 3 /3
1
2 3 /3
2
2
2 3 /3
2
2
2
2
2 3 /3
1
3
1
3 /3
0
3 /3
1
3
1.6 Trigonometric Functions
1.6 Trigonometric Functions
Even and Odd Trig Functions:
“Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change.
Cosine is an even function because: cos cos
Secant is also an even function, because it is the reciprocal of cosine.
Even functions are symmetric about the y - axis.
1.6 Trigonometric Functions
Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes.
Sine is an odd function because: sin sin
Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function.
Odd functions have origin symmetry.
1.6 Trigonometric Functions
1.6 Trigonometric Functions
Definition Periodic Function, Period
A function f(x) is periodic if there is a positive number p such that f(x + p) = f(x) for every value of x. The smallestsuch value of p is the period of p.
y a f b x c d
Vertical stretch or shrink;reflection about x-axis
Horizontal stretch or shrink;reflection about y-axis
Horizontal shift
Vertical shift
Positive c moves left.
Positive d moves up.is a stretch.1a
is a shrink.1b
1.6 Trigonometric Functions
2sinf x A x C D
B
Horizontal shift
Vertical shift
is the amplitude.
A
is the period.B
A
B
C
D 21.5sin 1 2
4y x
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions