1.1 Lines Increments If a particle moves from the point (x1,y1) to the point (x2,y2), the increments in its coordinates are

1.1 Lines Slope Let P1= (x1,y1) and P2= (x2,y2) be points on a nonvertical line L. The slope of L is P1(x1,y1) P2(x2,y2) Δy Q(x2,y1) 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. Proof: If L1 || L2, then θ1= θ2 and m1= m2. Conversely, if m1 = m2, then θ1= θ2 and L1 || L2. m1 m2 L1 L2 slope m1 slope m2 θ1 θ2 1

1.1 Lines Theorem: If two non vertical lines L1and L2 are perpendicular, then their slopes satisfy m1m2 = -1 and conversely. Proof: Δ ADC ~ ΔCDB L1 L2 Slope m2 Slope m1 A C B m1 = tan θ1 = a/h m2 = tan θ2 = -h/a h D a θ1 θ2 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 Analysis Plot the data Find the regression equation y = mx + b Superimpose the graph on the data points. Use the regression equation to predict y-values.

1.1 Lines Coordinate Proofs State given and prove. Draw a picture. Label coordinates, use (0,0) if possible. Fill in missing coordinates. Use algebra to prove parallel/perpendicular-slope equidistant-distance formula bisect-midpoint

1.1 Lines Prove the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. A(0,0) C(b,0) B(0,a) M(b/2,a/2) Given: ΔBAC is a right triangle Prove: AM = BM = CM Since AM = BM = CM, the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices

1.2 Functions and Graphs y = f(x) y is a function of x 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 a open a b closed a b half opened a b a open a b closed a b half opened a b half opened a b

1.2 Functions and Graphs 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 Domain [0, ∞) Range [0, ∞)

1.2 Functions and Graphs Function Domain Range y = x y = x2 y = |x| [-3,3] [0,3]

1.2 Functions and Graphs Definitions Even Function, Odd Function A function y = f(x) is an even 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 Domain Range (-,) Piece Functions [-3, )

1.2 Functions and Graphs Piece Functions Domain Range (-,) [0, )

1.2 Functions and Graphs Composite Functions f(g(x)) f(x) = x2, g(x) = 3x - 1 Find: f(g(2)) g(f(-1)) g(f(x)) 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 Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

1.3 Exponential Functions Use the rules for exponents to solve for x. 4x = 128 (2)2x = 27 2x = 7 x = 7/2 2x = 1/32 2x = 2-5 x = -5

1.3 Exponential Functions 27x = 9-x+1 (33)x = (32)-x+1 33x = 3-2x+2 3x = -2x+ 2 5x = 2 x = 2/5 (x3y2/3)1/2 x3/2y1/3

49 9 1/9 5 4 5 81 32 2 1 1/8 2 80/9 8 9/4 36 1/25 1/49 4 5

1.3 Exponential Functions Properties of f (x) = ax Domain: Range: Increasing for: Decreasing for: Point Shared On All Graphs: Asymptote: (-∞, ∞) (0, ∞) a > 1 0 < a < 1 (0, 1) y = 0

1.3 Exponential Functions Natural Exponential Function where e is the natural base and e  2.718…

1.3 Exponential Functions 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. y new amount yo original amount b base t time h 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. Find the amount remaining after t hours. Find the amount remaining after 60 hours. Estimate the amount remaining after 4 days. 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. Find the amount remaining after t hours. 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 Functions An 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 0.1 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 Find the amount after 2 weeks. 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

1.4 Parametric Equations 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 Equations defined in terms of x and y. These may or may not be functions. Some examples include: x2 + y2 = 4 y = x2 + 3x + 2

1.4 Parametric Equations Sketch the graph of the parametric equation for t in the interval [0,3] t x y 1 -1 3 2 -3 6 -5 9

1.4 Parametric Equations Eliminate the parameter t from the curve

1.4 Parametric Equations If we let t = the angle, then: Circle: Since: We could identify the parametric equations as a circle.

1.4 Parametric Equations Ellipse: 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.

How is t represented on this graph? 1.4 Parametric Equations How is t represented on this graph?

1.4 Parametric Equations   t =   t = 0 

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

(a) x = t2, y = t3 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. (a) x = t2, y = t3

1.4 Parametric Equations (b)

1.4 Parametric Equations (c) x = sec t, y = tan t where -/2 < t < /2 Hint: sec2 θ – tan2 θ = 1

1.4 Parametric Equations Find a parametrization for the line segment with endpoints (2,1) and (-4,5). x = 2 + at y = 1 + bt Cartesian Equation m = (5 – 1)/(-4 – 2) = -2/3 when t = 1, a = -6 when t = 1, b = 4 y = mx + b 1 = (-2/3)(2) + b b = 7/3 x = 2 – 6t and y = 1 + 4t 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.

1.5 Functions and Logarithms To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. one-to-one not one-to-one not a function (also not one-to-one)

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)

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 To prove functions are inverses show that f(f-1(x)) = f-1(f(x)) = x f −1(x) = y <=> f (y) = x

1.5 Functions and Logarithms To obtain the formula for the inverse of a function, do the following: Let f (x) = y. Exchange y and x. Solve for y. Let y = f −1(x).

1.5 Functions and Logarithms Given an x value, we can find a y value. Inverse functions: Switch x and y: Inverse functions are reflections about y = x. Solve for y:

1.5 Functions and Logarithms Prove f(x) and f-1(x) are inverses.

1.5 Functions and Logarithms Determine the formula for the inverse of the following one-to-one functions. (a) (b) (c)

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 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)

1.5 Functions and Logarithms Find the inverse of g(x) = 3x. Definition: x = a y if and only if y = log a x

1.5 Functions and Logarithms log a (ax) = x for all x   alog ax = x for all x > 0 log a (xy) = log a x + log a y log a (x/y) = log a x – log a y log a xn = n log a x Common Logarithm: log 10 x = log x Natural 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. 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? We use logs when we have an unknown exponent. 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): 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 Determine the exact value of log 8 2. Determine the exact value of ln e 2.3. Evaluate log 7.3 5 to four decimal places. Write as a single logarithm: ln x + 2ln y – 3ln z. Solve 2x + 5 = 3 for x.

1.6 Trigonometric Functions A B θ s r The Radian measure of angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle.

1.6 Trigonometric Functions θ terminal ray initial ray y x r P(x,y)

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

1.6 Trigonometric Functions 60° 1 2 30° 45° 1 S A T C

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: 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: Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry.

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 smallest such value of p is the period of p.

1.6 Trigonometric Functions Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. is a stretch. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. is a shrink.

1.6 Trigonometric Functions is the amplitude. Vertical shift is the period. Horizontal shift B A C D

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 Let  be the acute angle of a right triangle with sin  = 3/5. Find the exact values of the other five trig functions. Show all your work. 3  4 5

1.6 Trigonometric Functions If and find the exact value of  -1

1.6 Trigonometric Functions Find the amplitude, period, and frequency of the simple harmonic motion. Amplitude ¾ Frequency ¼

1.6 Trigonometric Functions Find the exact values without using a calculator: (a) tan (11/6) (b) sec(-3/4) (c) cot (-5/3)

1.6 Trigonometric Functions -3 -5  Given that tan  = 3/5,  in quadrant III, and cos  = -1/2,  in quadrant II, Find (a) cos( - ) (b) sin 2 (a) cos  cos  + sin  sin   -1 2 (b) sin 2 sin 2 = 2 sin  cos 

1.6 Trigonometric Functions Verify the identities. Show all your work. (a) (b) (c) sin(π + x) = -sin x sin  cos x + sin x cos  0 cos x + sin x (-1) = -sin x

1.6 Trigonometric Functions Find the exact values without using a calculator. (a) (b) (c) sec-1 (2) 120º 240º 150º 330º 60º 300º

1.6 Trigonometric Functions Find the exact values without a calculator. (c) tan(sec-1x) (a) (b)  5 3 4   1 cos 2 =cos2 - sin2

1.6 Trigonometric Functions Solve each equation for exact solutions in the interval [0,2). (a) cos2 x – 1 = 0 (b) 2 cos2 x + 1 = -3 cos x cos2 x = +1 2 cos2 x + 3 cos x+ 1 = 0 cos x = 1 or cos x = -1 (2cos x + 1)(cos x + 1) = 0 x = 0, x =  x = 2/3, 4/3 x =  sin2x = 0 2sin x cos x = 0 sin x = 0 or cos x = 0 x = 0, /2, ,3/2

1.6 Trigonometric Functions Solve each equation for exact solutions in the interval [0,2). (a) tan x sin x – sin x = 0 (b) 2cos x sin x - cos x = 0 sin x(tan x – 1) = 0 cos x(2sin x – 1) = 0 sin x= 0 or tan x – 1 = 0 cos x = 0 or 2 sin x – 1 = 0 x = 0,  x = /4, 5/4 x = /2, 3/2 or sin x = ½ x = /6, 5/6 (c) tan2 x = 3 x = /3, 2/3 4/3 5/3