AMHS PreCalculus 1. Unit 1. The Real Line. Interval notation is a convenient and compact way to express a set of numbers on the real number line.

Transcription

1 AMHS PreCalculus 1 Unit 1 The Real Line Interval Notation Interval notation is a convenient and compact way to express a set of numbers on the real number line. Graphic Representation Inequality Notation < x < 3 Interval notation 1 x 4 1< x x x < 1 Inequality Properties 1. If a> b, then a+ c> b+ c. If a> b and c < 0, then ac < bc Ex. 1 Solve each inequality (note that the degree is 1) and write the solution using interval notation: a) 3x + 5> 1 b) 9 x + 10< 5 c) 7 x 3< 4 3

2 AMHS PreCalculus Ex. Solve each inequality and write the solution using inequality notation. π π a) 0 x π b) 0 x π c) 0 x + π π 3 Polynomial Inequalities with degree two or more and Rational Inequalities Solve x 4x+ 7> 4 by making a sign chart. Write your answer using interval notation. 1. Set one side of the inequality equal to zero.. Temporarily convert the inequality to an equation. 3. Solve the equation for x. If the equation is a rational inequality, also determine the values of x where the expression is undefined (where the denominator equals zero). These are the partition values. 4. Plot these points on a number line, dividing the number line into intervals. 5. Choose a convenient test point in each interval. Only one test point per interval is needed. 6. Evaluate the polynomial at these test points and note whether they are positive or negative. 7. If the inequality in step 1 reads > 0, select the intervals where the test points are positive. If the inequality in step 1 reads < 0, select the intervals where the test points are negative.

3 AMHS PreCalculus 3 Ex. 3 Solve each inequality. Show the sign chart. Draw the solution on the number line and express the answer using interval notation. a) xx ( 4)( 3) + x 0 b) x 3x 4> 0 x 3 c) < 0 x 4 d) 3 x + 4 x 1

4 AMHS PreCalculus 4 Absolute Value x x if x 0 = x if x < 0 The absolute value of a real number x is the distance on the number line that x is from 0. Absolute value equations Ex. 4 Solve the equation (check your answers for extraneous solutions): a) x + 1 = 4 x 3 b) x + 3 = 1 Absolute value inequalities 1. if x < a, then a< x< a. if x > a 0, then x < a or x > a Ex. 5 Solve the inequality. Express your answers in interval notation and graph the solution: a) 4x 1 <.01 b) x 1 5

5 AMHS PreCalculus 5 c) Lines Equations and Graphs The equation y = mx+ b is a linear equation where m and b are constants. This is called Slope Intercept form where m is the slope and b is the y intercept. In general, m > 0 m < 0 m = 0 m is undefined

6 AMHS PreCalculus 6 The slope of a Line Point Slope equation of a line: Ex. 1 Find the point slope equation of a line passing through the points ( 1, ) and (,5). Ex. Write the equation of a line passing through the points (4,7) and (0,3).

7 AMHS PreCalculus 7 Parallel and Perpendicular Lines Two non vertical lines are parallel iff they have the same slope. Two lines with non zero slopes and m are perpendicular iff m1i m = 1. m1 Ex. 3 Find the equation of the line passing through the point ( 3,) that is parallel to 5x y =3. Ex. 4 Find the equation of the line passing through ( 4,3) which is perpendicular to the line passing through ( 3,) and (1,4). Ex. 5 A new car costs $9,000. Its useful lifetime is approximately 1 years, at which time it will be worth an estimated $ a) Find the linear equation that expresses the value of the car in terms of time. b) How much will the car be worth after 6.5 years?

8 AMHS PreCalculus 8 Ex. 6 The manager of a furniture factory finds that it costs $0 to manufacture 100 chairs and $4800 to manufacture 300 chairs. a) Assuming that the relationship between cost and the number of chairs produced is linear, find an equation that expresses the cost of the chairs in terms of the number of chairs produced. b) Using this equation, find the factory s fixed cost (i.e. the cost incurred when the number of chairs produced is 0). Ex. 7 Find the slope intercept equation of the line that has an x intercept of 3 and a y intercept of 4.

9 AMHS PreCalculus 9 Circles Recall the distance formula d = ( x x ) + ( y y ) 1 1 The Standard form for the equation of a circle is: Ex.1 Write the equation of a circle with center ( 1,) and radius 3. Sketch this circle. Ex. Write the equation of a circle with center at the origin and radius 1. Ex.3 Find the equation of the circle with center ( 4,1) that is tangent to the line x = 1.

10 AMHS PreCalculus 10 Ex. 4 Find the equation of the circle with center (4,3) and passing through the point (1,4). Ex. 5 Express the following equations of a circle in standard form. Identify the center and radius: a) x y 4x 6y + + =3 b) x x+ y + 4y = 4

11 AMHS PreCalculus 11 The intercepts of a graph The x coordinates of the x intercepts of the graph of an equation can be found by setting y = 0 and solving for x. The y coordinates of the y intercepts of the graph of an equation can be found by setting x = 0 and solving for y. Ex. 1 Find the x and y intercepts of the line and sketch its graph: x + y = 1 Ex. Find the x and y intercepts of the circle and sketch its graph: x + y = 9 Ex. 3 Find the intercepts of the graphs of the equations. a) x y = 9

12 AMHS PreCalculus 1 b) y x x = Symmetry In general : A graph is symmetric with respect to the y axis if whenever ( x, y) is on a graph ( x, y) is also a point on the graph. A graph is symmetric with respect to the x axis if whenever ( x, y) is on a graph ( x, y) is also a point on the graph. A graph is symmetric with respect to the origin if whenever ( x, y) is on a graph ( x, y) is also a point on the graph. Tests for Symmetry: The graph of an equation is symmetric with respect to: a) the y axis is replacing x by x results in an equivalent equation. b) the x axis is replacing y by y results in an equivalent equation. c) the origin if replacing x and y by x and y results in an equivalent equation. Ex. 1 Show that the equation y = x 3 has y axis symmetry.

13 AMHS PreCalculus 13 Ex. Show that the equation x+ y = 10 has x axis symmetry. Ex. 3 Show that the equation x + y = 9 has symmetry with respect to the origin. Ex. 4 Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the x axis, y axis, or origin. a) x = y b) y = x 4 c) y x x = d) y= x 9

14 AMHS PreCalculus 14 Algebra and Limits Difference of two squares: a b ( a b)( a b = + ) Difference of two cubes: a b = ( a b)( a + ab+ b 3 3 ) Sum of two cubes: a + b = ( a+ b)( a ab+ b 3 3 ) Binomial Expansion n = : ( a ± b) = a ± ab + b Binomial Expansion n = 3: ( a ± b) = a ± 3a b + 3ab ± b Limits x Ex. 1 Estimate lim x x 4 numerically by completing the following chart: x y x y x Conclusion: lim x x 4 = Properties of Limits If a and c are real numbers, then lim c= c,lim x= a,lim x n = a n x a x a x a Ex. Find the limit: a) 3 lim( x x+ 4) x b) lim (x + 7) x 1

15 AMHS PreCalculus 15 Ex. 3 Find the given limit by simplifying the expression a) x + x 6 lim x x 5x+ 6 b) lim x x x c) x + 3 lim x 1 x 1 d) x lim x 7x+ 10 x e) lim x x + x + 5 3

16 AMHS PreCalculus 16 Unit Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible inputs for the function. On a graph these are the values of the independent variable (most commonly known as the x values). The range is the set of all possible outputs for the function. On a graph these are the values of the dependent variable (most commonly known as the y values). We use the notation f ( x) to represent the value (again, in most cases, a y value) of a function at the given independent value of x. For any value of x, ( x, f ( x)) is a point on the graph of the function f ( x ). Ex. 1 Given f ( x) = x, graph the function and determine the domain and range. Use interval notation to express the domain and range.

17 AMHS PreCalculus 17 Ex. Given f ( x ) = to express the domain and range. x, graph the function and determine the domain and range. Use interval notation Ex. 3 For the function f( x) = x + x+ 4, find and simplify: a) f ( 3) b) f ( x+ h) Ex. 4 For x, x 0 f( x) = x 1, x< 0 find: a) f (1) = b) f ( 1) = c) f ( ) = d) f (3) =

18 AMHS PreCalculus 18 Ex. 5 The graph of the function f is given: a) Determine the values: f ( ) = f (0) = b) Determine the domain: c) Determine the range: f () = f (4) = Ex. 6 The graph of the function f is given: a) f ( 3) = f (0) = f (4) = b) For what numbers x is f( x ) = 0? c) What is the domain of f? d) What is the range of f? e) What is (are) the x intercept(s)? f) What is the y intercept? g) For what numbers x is f( x ) < 0? h) For what numbers x is f( x ) > 0?

19 AMHS PreCalculus 19 Vertical Line Test for a Function: An equation is a function iff every vertical line intersects the graph of the equation at most once. Ex. 7 Determine which of the curves are graphs of functions: a) b) c) Domain (revisited) Rule for functions containing even roots (square roots, 4 th roots, etc): Ex. 1 Determine the domain and range of f( x) = 4+ x 3 Ex. Determine the domain of f t t t () = + 15

20 AMHS PreCalculus 0 Rule for functions containing fractional expressions: 5x Ex. 3 Determine the domain of hx ( ) = x 3x 4 Ex. 4 Determine the domain of gx ( ) = x 1 + x 15 Ex. 5 Determine the domain of hx ( ) = 3 x x +

21 AMHS PreCalculus 1 Intercepts (revisited) The y intercept of the graph of a function is (0, f (0)). The x intercept(s) of the graph of a function f ( x) is/are the solution(s) to the equation f ( x ) = 0. These x values are called the zeros of the function f ( x ). Ex. 1 Find the zeros of f( x) = x(3x 1)( x+ 9) Ex. Find the zeros of f x x x ( ) = Ex. 3 Find the zeros of f x 4 ( ) = x 1 Ex. 4 Find the x and y intercepts (if any) of the graph of the function 1 f( x) = x 4

22 AMHS PreCalculus Ex. 5 Find the x and y intercepts (if any) of the graph of the function f x ( ) = 4( x ) 1 Ex. 6 Find the x and y intercepts (if any) of the graph of the function f( x) = x + 4 x 16 Ex. 7 Find the x and y intercepts (if any) of the graph of the function 3 f ( x) = 4 x

23 AMHS PreCalculus 3 Transformations Horizontal and Vertical shifts Suppose y = f( x) is a function and c is a positive constant. Then the graph of 1. y = f( x) + cis the graph of f shifted vertically up c units.. y = f( x) c is the graph of f shifted vertically down c units. 3. y = f( x+ c) is the graph of f shifted horizontally to the left c units. 4. y = f( x c) is the graph of f shifted horizontally to the right c units. Ex. 1 Consider the graph of a function y = f( x) y = f( x) + 3 y = f( x) y = f( x 1) y = f( x+ 3)

24 AMHS PreCalculus 4 Suppose y = f( x) is a function. Then the graph of 1. y = f( x) is the graph of f reflected over the x axis.. y = f ( x) is the graph of f reflected over the y axis. Ex. Consider the graph of a function y = f( x) Sketch y = f( x ) + 3 Common Functions f ( x) = x f ( x) = x

25 AMHS PreCalculus 5 f ( x) = x f ( x) = 3 x f ( x) 3 = x 1 f( x) = x f ( x) = [ x]

26 AMHS PreCalculus 6 Combining common functions with transformations Sketch the graphs of the following functions. Determine the domain and range and any intercepts. Ex. 1 f( x) = x+ + 1 Ex. f( x) = 1 x Ex. 3 f x 3 ( ) ( x ) 1 = Ex. 4 f( x) = x 1+ 3

27 AMHS PreCalculus 7 Symmetry (revisited) Tests for Symmetry The graph of a function f is symmetric with respect to: 1. the y axis if f ( x) = f( x) for every x in the domain of the f ( x ).. the x axis if f ( x) = f( x) for every x in the domain of the f ( x ). If the graph of a function is symmetric with respect to the y axis, we say that f is an even function. If the graph of a function is symmetric with respect to the origin, we say that f is an odd function. In examples 1 3, determine whether the given function y = f( x) is even, odd or neither. Do not graph. Ex. 1 f ( x) x x 5 3 = + +x Ex. f( x) = x + 3 Ex. 3 f x ( ) x = + x

28 AMHS PreCalculus 8 Transformations Vertical Stretches and Compressions Suppose y = f( x) is a function and c a positive constant. The graph of y = cf( x) is the graph of f 1. Vertically stretched by a factor of c if c > 1. Vertically compressed by a factor of c if 0< c < 1 Ex.1 Given the graph of y = f( x) a) Sketch y = f ( x) b) 1 y= f ( x ) Ex. Sketch the graph of the following functions. Include any intercepts. f ( x) = x 1 f( x) = 3( x 1)

29 AMHS PreCalculus 9 Quadratic Functions A quadratic function constants. y = f( x) is a function of the form = + + where a 0, b and c are f ( x) ax bx c The graph of any quadratic function is called a parabola. The graph opens upward if a > 0 and downward if a < 0. The domain of a quadratic function is the set of real numbers (, ). A quadratic function has a vertex (which serves as the minimum or maximum of the function depending on the value of a ), a line of symmetry, and may have zero, one or two x intercepts. Ex. 1 Sketch the graph of f( x) = ( x 1) + 3. Determine any intercepts.

30 AMHS PreCalculus 30 The standard form of a quadratic function is parabola and x = h is the line of symmetry. f ( x) a( x h) k = + where ( hk), is the vertex of the Ex. Rewrite the quadratic function f ( x) = x x 3in standard form by completing the square. Determine any intercepts, the vertex, the line of symmetry and sketch the graph. Ex. 3 Rewrite the quadratic function f ( x) = 4x + 1x 9in standard form by completing the square. Determine any intercepts, the vertex, the line of symmetry and sketch the graph.

31 AMHS PreCalculus 31 Ex. 4 Comple the square to find all the solutions to the equation ax bx c + + = 0 The vertex of any parabola of the form = + + is ( b b, f ( a a )). f ( x) ax bx c Ex. 5 Find the vertex of the quadratics from examples and 3 directly by using ( b b, f ( a a )). Ex. 6 Find the vertex from example by using the x intercepts and the line of symmetry.

32 AMHS PreCalculus 3 Ex.7 Find the intercepts and vertex of the function 1 f x = x + x+ ( ) 1 Ex. 8 Find the maximum or the minimum of the function. 1. f x x x ( ) = f x x x ( ) = Ex.9 Determine the quadratic function whose graph is given.

33 AMHS PreCalculus 33 Freely Falling Object Suppose an object, such as a ball, is either thrown straight upward or downward with an initial velocity v0 or simply dropped ( v 0 = 0 ) from an initial height s 0. Its height, st () as a 1 function of time t can be described by the quadratic function st () = gt + vt+ s Gravity on earth is 3 ft / sec or 9.8 m / sec. Also, the velocity of the object while it is in the air is v() t = gt + v0 0 0 Ex. 10 An arrow is shot vertically upward with an initial velocity of the ground. 64 ft / sec from a point 6 feet above 1. Find the height st () and the velocity vt () of the arrow at time t 0.. What is the maximum height attained by the arrow? What is the velocity of the arrow at the time it attains its maximum height? 3. At what time does the arrow fall back to the 6 foot level? What is its velocity at this time? Ex. 11 The height above the ground of a toy rocket launched upward from the top of a building is given by st ( ) = 16t + 96t What is the height of the building?. What is the maximum height attained by the rocket? 3. Find the time when the rocket strikes the ground. What is the velocity at this time?

34 AMHS PreCalculus 34 Silly String Activity Unit 3 Objective: The use a quadratic function to model the path of silly string. Materials: Can of silly string, tape measure, stopwatch, clear overhead transparency, TI84 Personnel: Timekeeper, Silly String operator, assistant Calculate the initial velocity of the silly string as it exits the can. v 0 1. Hold the can of silly string 1 foot above the ground. Have the timekeeper start the stopwatch and say go. At this time, shoot a short burst of silly string towards the ceiling. Have the class keep a casual eye on the maximum height the silly string achieves. When the silly string hits the floor, have the timekeeper stop the stopwatch and record the elapsed time.. Measure the maximum height of the silly string observed by the class. Use the position equation 1 st () = gt + vt 0 + s 0 with g = 3 ft / sec to calculate. ( s = 1, get t from the timekeeper. This represents the time it took for the silly v0 0 string to reach the ground, i.e. st () =0) Now that we know gv, 0 and s0 we can set up a position equation to model the height of the silly string as a function of time. Use this equation to determine the maximum height (the vertex holds this info) of the silly string. How does this compare to the actual height observed by the class. What factors might have caused it to be different? Now we are going to get the assistant to lean over the can of silly string (with the clear overhead transparency protecting the face) in its original position 1 foot above the ground and see if the assistant can move fast enough to avoid getting silly string in the face. Calculate the time it would take for the silly string to reach the assistant s face (set st () = the height of the assistant s face and solve fort ) Once the reaction time for the assistant has been calculated and discussed, see if the assistant can actually react that quickly, i.e. avoid silly string in the face. To date, it has never been done. Enjoy!

35 AMHS PreCalculus 35 Horizontal Stretches and Compressions Suppose y = f( x) is a function and c a positive constant. The graph of y = f( cx) is the graph of f 1. Horizontally compressed by a factor of 1 c if c > 1. Horizontally stretched by a factor of 1 c if 0< c < 1 Ex.1 Given the graph of y = f( x) c) Sketch y = f ( x) d) 1 y= f( x) Ex. Consider the function f x ( ) = x 4 a) On the same axis, sketch f ( x), f ( x) and 1 f ( x ). Identify any intercepts of each function.

36 AMHS PreCalculus 36 b) On the same axis, sketch ( ), ( ) f x f x and 1 f ( x ). Identify any intercepts of each function. c) List the transformations on f ( x) = x required to sketch f( x) = x+ 1 Piecewise Defined Functions A function f may involve two or more functions, with each function defined on different parts of the domain of f. A function defined in this manner is called a piecewise defined function. Ex.1 Sketch the graph of the given function and find the following: x if x 0 f( x) = x = x if x < 0 a) f ( 1) = b) f () = c) Domain: d) Range: Ex. 1b Express f( x) = x+ 3 as a piecewise function:

37 AMHS PreCalculus 37 Ex. Sketch the graph of the given function and find the following: 3x+ 1 if x 1 f( x) = x 1 if x> 0 a) f ( 1) = b) f () = c) Domain: d) Range: Ex.3 Graph the following a) 1 if x < 0 f( x) = 0 if x= 0 x + 1 if x > 0 b) f( x) = x 1 x 1 Hint: write this as a piecewise function Domain: Range: Domain: Range:

38 AMHS PreCalculus 38 Graphing the Absolute Value of a Function Sketch the graph of the given functions. Include any intercepts. Ex.1 x if x 0 f( x) = x = x if x < 0 Ex. f( x) = ( x ) + 4 Ex.3 Ex.4 f( x) = x 4x+ 3 f( x) = x+ 3 1

39 AMHS PreCalculus 39 Compositions of Functions The composition of the function f with the function g, denoted f g is defined by ( f g)( x) = f( g( x)). The domain of f g consists of those x values in the domain of g for which g( x) is in the domain of f. Ex. 1 f( x) = x and gx ( ) = x 1. Find the following: a) ( f g)( x) b) Find the domain of ( f g)( x) c) ( g f)( x) d) Find the domain of ( g f )( x) e) ( f g)( ) f) ( g f)( 4) g) ( g f)( 1) Ex. Write the function f x ( ) x 3 = + as the composition of two functions Ex.3 Write the function f( x) = x 3 as the composition of three functions. 4x+ 1

40 AMHS PreCalculus 40 Ex. 4 Given F( x) = ( x+ 4) x+ 4 find functions f and g such that F( x) = ( f g)( x). Ex. 5 A metal sphere is heated so that t seconds after the heat had been applied, the radius by rt () = tcm. Express the Volume of the sphere as a function of t. rt () is given Ex. 6 f ( x) = x and gx ( ) = x( x 0). Find the following: a) ( f g)( x) b) ( g f)( x) c) g(3) d) g(4) e) f (9) f) f (16)

41 AMHS PreCalculus 41 Inverse Functions Suppose that f is a one to one function with domain X and range Y. The inverse function for the function f is the function denoted f 1 ( f( x)) = xand f f x 1 ( ( )) = x. 1 f with domain Y and range X and defined for all values x X by Ex. 1 Prove that f( x) = x and gx ( ) x = + ( x 0)are inverse functions using composition. Steps for Finding the Inverse of a Function: 1. Set y = f( x). Change x y and y x 3. Solve for y 4. Set y = f 1 ( x) Ex. Find the inverse of f ( x) and f 1 ( x). 3x f( x) = and check using composition. Find the domain and range of ( x 4)

42 AMHS PreCalculus 4 The graph of f 1 ( x) is a reflection of the graph of f ( x) about the line y = x. One to One Functions A function is one to one iff each number in the range of f is associated with exactly one number in its domain. In other words, f ( x1) = f( x ) implies x 1 = x. Horizontal Line Test for One to One Functions A function is one to one precisely when every horizontal line intersects its graph at most once. Ex. 3 Determine whether the given function is one to one a) f x 3 ( ) x = + b) f ( x) = x x

43 AMHS PreCalculus 43 Ex.4 Given f( x) = x 3 Domain of f ( x ): Range of f ( x ): Domain of f Range of f 1 ( x) : 1 ( x) : Find f 1 ( x) and check using composition. Sketch the graph of f 1 ( x) and f ( x) on the same axis.

44 AMHS PreCalculus 44 Translating Words into Functions In calculus there will be several instances where you will be expected to translate the words that describe a problem into mathematical symbols and then set up or construct an equation or a function. In this section, we will focus on problems that involve functions. We begin with a verbal description about the product of two numbers. Ex.1 The sum of two nonnegative numbers is 15. Express the product of one and the square of the other as a function of one of the numbers. Ex. A rectangle has an area of 400 in length of one of its sides.. Express the perimeter of the rectangle as a function of the Ex.3 Express the area of a circle as a function of its diameter d.

45 AMHS PreCalculus 45 Ex. 4 An open box is made from a rectangular piece of cardboard that measures 30 cm by 40cm by cutting a square of length x from each corner and bending up the sides. Express the volume of the box as a function of x. Ex. 5 Express the area of the rectangle as a function of x. The equation of the line is x+ y =4.The lower left hand corner is on the origin and upper right hand corner of the rectangle with coordinate ( x, y) is on the line. Ex. 6 Express the area of an equilateral triangle as a function of the length s of one of its sides.

46 AMHS PreCalculus 46 The Tangent Line Problem Find a tangent line to the graph of a function f. m tan = lim Δx 0 f ( a+ Δx) f( a) Δx Ex.1 Find the slope of the tangent line to the graph of f x = + at x = 1. ( ) x Ex. Find the slope of the tangent line to the graph of f x = + at x = 3. ( ) x

47 AMHS PreCalculus 47 Ex.3 Find the slope of the tangent line to the graph of f x = + at x =. ( ) x The derivative of a function y = f( x) is the function f ' defined by: f '( x) = f ( x+δx) f( x) lim Δ x 0 Δx Ex.4 Find the derivative of f( x) = x +. Ex.5 Find the derivative of tangent line at x =. f ( x) = x 6x+ 3 and use it to find the slope and then the equation of the

48 AMHS PreCalculus 48 Ex. 6 Find the slope of the tangent line to the graph of f( x) = at x = 1. x Ex.7 Find the derivative of line at x =. f( x) = and use it to find the slope and then the equation of the tangent x Ex. 8 Find the derivative of f( x) = x.

49 AMHS PreCalculus 49 Polynomial Functions Unit 4 Polynomial and Rational Functions A polynomial function y = p( x) is a function of the form px ( ) ax a x a x... ax ax a n n 1 n = n + n 1 + n where an, an,..., a, a, a are real constants and are called the coefficients of px ( ) n is the degree of px ( ) and is a positive integer. is called the leading coefficient and a is the an 0 constant term of the polynomial. The domain of any polynomial is all real numbers. Ex. 1 Determine the degree, the leading coefficient and the constant term of the polynomial. a) 4 3 f( x) = 5x + 7x 3x+ 7 b) 3 gx ( ) = 13x+ 5x 4x End Behavior There are four scenarios: 4 1) Sketch p( x) = x, p( x) = x ( n is even, a > 0 ) n ) Sketch p( x) = x, p( x) = x even, < 0 ) a n 4 ( n is As x, p( x) As x, p( x) As x, p( x) As x, p( x)

50 AMHS PreCalculus ) Sketch p( x) = x, p( x) = x ( n is odd, a > 0 ) n 3 5 4) Sketch p( x) = x, p( x) = x ( n is odd, a < 0 ) n As x, p( x) x, p( x) As x, p( x) x, p( x) As x and x, the graph of the polynomial n n 1 n n px ( ) = anx + an 1x + an x ax + ax 1 + a 0 resembles the graph of y = anx. Ex. Use the zeros and the end behavior of the polynomial to sketch an approximation of the graph of the function. a) f x 3 ( ) x 9 = x b) gx x x 4 ( ) = 5 + 4

51 AMHS PreCalculus 51 c) f ( x) 5 = x +x Repeated Zeros k If a polynomial f ( x) has a factor of the form ( x c), where k > 1, then x = c is a repeated zero of multiplicity k. If If k is even, the graph of f ( x) flattens and just touches the x axis at x = c. k is odd, the graph of f ( x) flattens and crosses the x axis at x = c. Ex. 4 Sketch the given graphs f x x x x 4 3 ( ) = 3 + gx x x x 3 ( ) = ( 1) ( + )( 3)

52 AMHS PreCalculus 5 Ex. 5 The cubic polynomial px ( ) has a zero of multiplicity two at x = 1 and a zero of multiplicity one at x =. Also p( 1) =. Determine px ( ) and sketch the graph. Ex. 6 An open box is to be made from a rectangular piece of cardboard that is 1 by 6 feet by cutting out squares of side length x ft from each corner and folding up the sides. a) Express the volume of the box vx ( ) as a function of the size x cut out at each corner. b) Use your calculator to approximate the value of x which will maximize the volume of the box. Ex. 7 The product of two non negative numbers is 60. What is the minimum sum of the two numbers?

53 AMHS PreCalculus 53 The Intermediate Value Theorem Suppose that f is continuous on the closed interval [ ab, ] and let N be any number between f ( a) and f () b, where f ( a) f( b). Then there exists a number c in ( ab, ) such that f () c = N. Ex. 1 Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f ( x) = x + x 1, [0,5], f() c = 11 Ex. Show that there is a root of the equation 3 x x = in the interval (0,1).

54 AMHS PreCalculus 54 The Division Algorithm Let f ( x) and dx ( ) 0be polynomials where the degree of f ( x) of dx ( ). Then there exists unique polynomials qx ( ) and rx ( ) such that f ( x) r( x) = qx ( ) + or f ( x) = d( x) q( x) + r( x). d( x) d( x) where rx ( ) has a degree less than the degree of dx ( ). is greater than or equal to the degree Ex 1. Divide the given polynomials. a) x x + x 4 x b) 3 x 1 x 1 c) 3 3 x x x+ 6 x + 1

55 AMHS PreCalculus 55 Remainder Theorem If a polynomial f ( x) is divided by a linear polynomial x c, then the remainder r is the value of f ( x) at x = c. In other words, f () c = r Ex. Use the Remainder Theorem to find r when 3 f( x) = 4x x + 4is divided by x. Ex. 3 Use the Remainder Theorem to find f () c for 4 f( x) = 3x 5x + 7when 1 c = Synthetic Division Synthetic division is a shorthand method of dividing a polynomial px ( ) by a linear polynomial x c. It uses only the coefficients of px ( ) and must include all 0 coefficients of px ( ) as well.

56 AMHS PreCalculus 56 Ex. 4 Use synthetic division to find the quotient and remainder when 3 a) f( x) = x 1is divided by x 1 b) 4 f ( x) = x 14x + 5x 9is divided by x + 4 c) 4 3 8x 30x 3x 8x is divided by 1 x 4 Ex. 5 Use synthetic division and the Remainder Theorem to find f () c for f( x) = 3x + 4x + x 8x 6x + 9 when c =.

57 AMHS PreCalculus 57 Ex. 6 Use synthetic division and the Remainder Theorem to find f () c for 3 f( x) = x 7x + 13x 15 when c = 5. The Factor Theorem A number c is a zero of a polynomial px ( ) ( pc () = 0) if and only if ( x c) is a factor of px ( ). Ex. 1 Determine whether a) x + 1is a factor of f x x x x 4 ( ) = b) x is a factor of x 3x Fundamental Theorem of Algebra A polynomial function px ( ) of degree n > 0 has at least one zero. In fact, every polynomial function px ( ) of degree n > 0 has at exactly n zeros.

58 AMHS PreCalculus 58 Complete Factorization Theorem Let c1, c,... c n be the n (not necessary distinct) zeros of the polynomial function of degree n > 0 : px ( ) = ax + a x + a x ax + ax+ a 0. n n 1 n n n 1 n 1 Then px ( ) can be written as the product of n linear factors p( x) = a ( x c )( x c ) ( x c ). n 1 n Ex.1 Give the complete factorization of the given polynomial px ( ) with given information: a) 3 p( x) = x 9x + 6x 1; 1 x = is a zero. 4 3 b) px ( ) = 4x 8x 61x + x+ 15; x= 3, x= 5 are both zeros.

59 AMHS PreCalculus 59 3 c) px ( ) = x 6x 16x+ 48; ( x ) is a factor. d) 4 3 = + x x(3x 1) p( x) 3x 7x 5x ; is a factor. Ex. Find a polynomial function f ( x) of degree three, with zeros 1, 4, 5 such that the graph possesses the y intercept (0,5).

60 AMHS PreCalculus 60 The Rational Zero Test Suppose p q is a rational zero of 1 ( ) n n f x a x a x a x n = a x + a x+ a, n n 1 n 1 0 where p 0 a0, a1..., an are integers and an 0. Then divides a and q divides an. The Rational Zero Test provides a list of possible rational zeros. Ex. 1 Find all the rational zeros of f ( x) then factor the polynomial completely. a) f x x x x x 4 3 ( ) = b) f x x x x x 4 3 ( ) =

61 AMHS PreCalculus 61 Complex Roots of Polynomials Consider factoring the function: f x 3 ( ) = x 1 The Square Root of 1 We define i = 1 so that i = 1. Complex Numbers A complex number is a number of the form a+ biwhere a and b are real numbers. The number a is called the real part and the number b is called the imaginary part. Complex Arithmetic Ex. 1 a) (+ 3 i) (6 i) = b) (+ 3 i)(4 i) = c) (3 6 i)(3+ 6 i) d) (4 5)(4 i + 5) i Complex Conjugates The complex conjugate for a complex number z = a+ biis z = a bi. In general, ( a bi)( a+ bi) =

62 AMHS PreCalculus 6 Ex. a) ( + 3 i) = (1 6 i) ( i) = (1+ 7 i) Ex. 3 Simplify: a) 4 b) 8 Ex. 4 Determine all solutions to the equation x 4x+ 13= 0 Ex. 5 Completely factor f x 3 ( ) = x 1.

63 AMHS PreCalculus 63 Ex. 6 Find the complete factorization of multiplicity two. 4 3 f( x) = x 1x + 47x 6x+ 6 given that 1 is a zero of Conjugate Pairs of Zeros of Real Polynomials If the complex number z = a+ biis a zero of some polynomial px ( ) with real coefficients, then its conjugate z = a biis also a zero of px ( ). Ex. 7 Find a 3 rd degree polynomial gx ( ) with real coefficients and a leading coefficient of 1 with zeros 1 and 1 i. 4 3 Ex i is a zero of f( x) = x x 4x + 18x 45. Find all other zeros and then give the complete factorization of f ( x ).

64 AMHS PreCalculus 64 Rational Functions A rational function functions. y= f( x) is a function of the form p( x) f( x) =, where p and q are polynomial qx ( ) Ex. 1 Recall the parent function f( x) 1 =. Use transformations to sketch x gx ( ) = x 1 Asymptotes of Rational Functions The line x = a is a vertical asymptote of the graph of f ( x) if f( x) or f( x) as x a + (from the right) or x The graph of a (from the left). p( x) f( x) = has vertical asymptotes at the zeros of qx ( ) after all of the common factors qx ( ) of px ( ) and qx ( ) have been canceled out; the values of x where qx ( ) = 0and px ( ) 0. The graph of p( x) f( x) = has a hole at the values of x where qx ( ) = 0and px ( ) = 0. qx ( )

65 AMHS PreCalculus 65 The line y= bis a horizontal asymptote of the graph of f ( x) if f ( x) bwhen x or x. In particular, with a rational function n px ( ) ax n + a x ax+ a0 f( x) = = m qx ( ) bx + b x bx+ b m n 1 n 1 1 m 1 m There are three cases: a) If n< m, then y = 0is the horizontal asymptote. f( x) = 3x x x b) If n m, then = n a y = is the horizontal asymptote. b m f( x) = 3 3x + 6x 3 4x + x 3 c) If n> m, then there is no horizontal asymptote x x 5x + 1 f( x) = 3x 4x Slant Asymptote If the degree of numerator is exactly one more than the degree of the denominator, the graph of f ( x) has a slant asymptote of the form y= mx+ b. The slant asymptote is the linear quotient found by dividing px ( ) by qx ( ) and essentially disregarding the remainder. f( x) = x 3 8x+ 1 x + 1

66 AMHS PreCalculus 66 Ex. Find all asymptotes and intercepts and sketch the graphs of the given rational functions: a) f( x) = x 1 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x intercepts: y intercept: b) 3x + f( x) = x + 4 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x intercepts: y intercept:

67 AMHS PreCalculus 67 c) x + 3 f( x) = ( x )( x+ 5) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x intercepts: y intercept: d) f( x) = x x x + 1 Domain: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x intercepts: y intercept:

68 AMHS PreCalculus 68 x + 1 e) f( x) = x x Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x intercepts: y intercept: f) (3x+ 1)( x ) f( x) = ( x )( x+ 1) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x intercepts: y intercept: Ex. 3 Sketch the graph of a rational function that satisfies all of the following conditions: + f( x) as x 1 and f( x) as x 1 f( x) as x and f( x) as x f ( x) has a horizontal asymptote y = 0 f ( x) has no x intercepts Has a local maximum at ( 1, ) +

69 AMHS PreCalculus 69 Mandelbrot Set Activity using Fractint fractal generator Honors Precalculus Academic Magnet High School STEP 1 CREATE, SAVE, and PRINT an inspirational, visually pleasing area of the Mandelbrot set. Important Menu Items: VIEW Image Settings, Zoom In/ Out box, Coordinate Box FRACTALS Fractal Formula, Basic Options, Fractal Parameters COLORS Load Color Map Save As FILE 1) Start Fractint by clicking on the desktop icon. Fractint always starts with the Mandelbrot set, but in case things get weird, ALWAYS make sure mandel is selected in the Fractal Fractal Formula menu item. Use the Image Settings box to set the size of the picture (800 x 600 should work fine). ) Use the Zoom In/Out feature along with the Colors Load Color Map to create a variation of the Mandelbrot set. If the color palettes do not load, double click on the box that is labeled Pallette Files (*.Map) If you zoom in a few times you lose detail, you can increase the iterations in the Fractals Basic Options Box Remember that the more iterations the computer has to perform, the longer it will take 3) Use the Fractals Fractal Params window to record the x and y mins and maxs of the viewing rectangle on the imaginary plane. 4) Using the Coordinates box, point your arrow to a point you think is in the Mandelbrot set and record the x and y values. 5) Repeat #4 for a point you think is not in the set. 6) SAVE the fractal. Write down the coordinates (x and y mins and maxs) and number of iterations of your current position in the Mandelbrot set. 7) Print your fractal. STEP Create a typed text document (1 page or so) including, but not limited to: STEP 3 Typed: The NAME of your group s fractal and the name of everyone in your group A short story about your creation (what it makes you think of, color, choice, etc.) 1) List the x and y mins and maxs for your viewing rectangle from Step 1 ) Recall the coordinates of the point you thought was in the Mandelbrot set from Step 1. Let x = a and y = b for the complex number a + bi Let this number a + bi = c iterate this value 100 or more times using the Mandelbrot sequence:

70 AMHS PreCalculus 70 x 0 = c x 1 = x 0 + c x = x 1 + c Etc You will be using decimals and your calculator. Unlike the fractals, these calculations will not be pretty. Let your TI 84 do the work for you (i is above the decimal point.) 3) Record the last 0 iterations for analysis. Remember that you may need to scroll the TI 84 to the right to get the entire number 4) Were your predictions right about this point? Do you need more information to determine if it is in the set? 5) Repeat for the point you thought was not in the set, 6) Summarize your findings. TURN IN ALL 3 STEPS PAPERCLIPPED together in order. Extra Credit: Create your own color map. for info on Fractint

71 AMHS PreCalculus 71 Unit 5 Graphs of Functions Revisited Solving Equations Graphically The Intersection Method To solve an equation of the form f ( x) = g( x) : 1. Graph y1 = f( x) and y = gx ( ) on the same screen.. Find the x coordinate of each point of intersection. Ex. 1 Solve: a) x + 1 = 4 x 3 b) x = 3x + 4 c) 3 x x x x 4 3 = + 6 The x intercept Method To solve an equation of the form f ( x) = g( x) : Ex. Solve: 1. Write the equation in the equivalent form f( x ) = 0.. Graph y = f( x). 3. The x intercepts of the graph are the real solutions to the equation. a) x + 1 = 4 x 3 b) x = 3x + 4 c) 3 x x x x 4 3 = + 6 d) 5 3 x + x = x + 5

72 AMHS PreCalculus 7 Technological Quirks 1. Solve f( x ) = 0 by solving f( x ) = 0.. Solve Ex. 3 Solve: f( x) = 0 by solving f( x ) = 0 (eliminate any values that also make ). gx ( ) g( x ) = 0 a) 4 x x x + 1= 0 b) x + x 1 9x 9x+ = 0 Applications Ex. 1 According to data from the U.S. Bureau of the Census, the approximate population of Chicago and Los Angeles between 1950 and 000 are given by: y (in millions) Chicago: y = x x + x Los Angeles: 3 y = x x x where x = 0 corresponds to In what year did the two cities have the same population? Ex. The average of two real numbers is 41.15, and their product is Find the two numbers. Ex. 3 A rectangle is twice as wide as it is high. If it has an area of 4.5 square inches, what are its dimensions?

73 AMHS PreCalculus 73 3 Ex. 4 A rectangular box with a square base and no top is to have a volume of 30,000 cm. If the surface area of the box is 6000 cm, what are the dimensions of the box? Ex. 5 A box with no top that has a volume of 1000 cubic inches is to be constructed from a x 30 inch sheet of cardboard by cutting squares of equal size from each corner and folding up the sides. What size square should be cut from each corner? Ex. 6 A pilot wants to make 840 mile trip from Cleveland to Peoria and back in 5 hours flying time. There will be a headwind of 30 mph going to Peoria, and it is estimated that there will be a 40 tail wind on the return trip. At what constant engine speed should the plane be flown?

74 AMHS PreCalculus 74 Solving Inequalities Graphically 1. Rewrite the inequality in the form f ( x ) < 0or f( x ) > 0.. Determine the zeros of f. 3. Determine the interval(s) where the graph is above ( f( x ) > 0) or below ( f ( x ) < 0) the x axis. Ex. 1 Solve each inequality graphically. Express your answer in interval notation. a) xx ( 4)( 3) + x 0 b) x 3x > 4 x 3 c) < 0 x 4 d) 3 x + 4 x 1 e) f) 4 3 x 6x + x < 5x Ex. A company store has determined the cost of ordering and storing x laser printers is: 300,000 c = x+ x If the delivery truck can bring at most 450 printers per order, how many printers should be ordered at a time to keep the cost below $ ?

75 AMHS PreCalculus 75 Increasing, Decreasing and Constant Functions A function f is increasing on an interval when, for any x1 and x in the interval, x 1 < x implies f ( x ) < f( x ). 1 A function f is decreasing on an interval when, for any x1 and x in the interval, x 1 < x implies f ( x ) > f( x ). 1 A function f is decreasing on an interval when, for any x1 and x in the interval, f ( x1) = f( x). Ex.1 Determine the open intervals on which each function is increasing, decreasing or constant. a) f( x) = x 1+ x 3 b) 3 f ( x) x 3x = c) f ( x) = x 3

76 AMHS PreCalculus 76 Relative Minimum and Maximum Values (Relative Extrema) A function value f ( a) is called relative minimum of f when there exists an interval ( x1, x) that contains a such that x1 < x< ximplies f ( a) f( x). A function value f ( a) is called relative maximum of f when there exists an interval ( x1, x) that contains a such that x1 < x< ximplies f ( a) f( x ). Ex. Determine the relative minimum and x intercepts of f x x x ( ) = 3 4 Ex. 3 Use a graphing utility to determine the relative minimum and x intercepts of f x x x ( ) = 3 4 Ex. 4 Use a graphing utility to determine any relative minima or maxima for f ( x) 3 = x +x

77 AMHS PreCalculus 77 Ex. 5 During a 4 hour period, the temperature tx ( ) (in degrees Fahrenheit) of a certain city can be 3 approximated by the model tx ( ) =.06x 1.03x + 10.x+ 34, 0 x 4 where x represents the time of day, with x = 0 corresponding to 6 A.M. Approximate the maximum and minimum temperatures during this 4 hour period. Optimization Translating Words into Functions revisited Ex.1 The sum of two nonnegative numbers is 15. Express the product of one and the square of the other as a function of one of the numbers. Use a graphing utility to find the maximum product. Ex. A rectangle has an area of 400 in. Express the perimeter of the rectangle as a function of the length of one of its sides. Use a graphing utility to find the minimum perimeter. Ex. 3 An open box is made from a rectangular piece of cardboard that measures 30 cm by 40cm by cutting a square of length x from each corner and bending up the sides. Express the volume of the box as a function of x. Use a graphing utility to find the dimensions of the box with the maximum volume.

78 AMHS PreCalculus 78 Ex. 4 Express the area of the rectangle as a function of x. The equation of the line is x+ y =4.The lower left hand corner is on the origin and upper right hand corner of the rectangle with coordinate ( x, y) is on the line. Use a graphing utility to find the rectangle with the maximum area. Concavity and Inflection Points Concavity is used to describe the way a curve bends. For any two points in a given interval that lie on a curve, if the line segment that connects them is above the curve, then the curve is said to be concave up over the given interval. If the segment is below the curve, then the curve is said to be concave down over the interval. A point where the curve changes concavity is called an inflection point.

79 AMHS PreCalculus 79 Ex. 1 For the following functions, estimate the following: 1. All local maxima and minima (relative extrema) of the function. Intervals where the function is increasing and/or decreasing 3. All inflection points of the function 4. Intervals where the function is concave up and when it is concave down a) 3 f ( x) = x + 6x x+ 3 b) gx x x 3 ( ) = c) f( x) = ( x ) d) f( x) = x x x + 1

80 AMHS PreCalculus 80 Right Triangle Trigonometry Unit 6 Trigonometry Part1 a) sin( α ) = d) csc( α ) = Hypotenuse b) cos( α ) = e) sec( α ) = α Opposite c) tan( α ) = f) cot( α ) = Adjacent Ex. 1 Find the values of the six trigonometric functions of the angleθ. 7 3 θ Ex. Find the exact values of the sin,cos, and tan of 45

81 AMHS PreCalculus 81 Ex. 3 Find the exact values of the sin,cos, and tan of 60 and 30 Ex. 4 Find the exact value of x (without calculator). 30 x 5 Ex. 5 Find all missing sides and angles (with calculator). 33 1

82 AMHS PreCalculus 8 Applications An angle of elevation and an angle of depression can be measured from a point of reference and a horizontal line. Draw two figures to illustrate. Ex. 6 (use a calculator) A surveyor is standing 50 feet from the base of a large building. The surveyor measures the angle of elevation to the top of the building to be How tall is the building? Draw a picture. Ex. 7 A ladder leaning against a house forms a 67 angle with the ground and needs to reach a window 17 feet above the ground. How long must the ladder be?

83 AMHS PreCalculus 83 Angles Degrees and Radians. An angle consists of two rays that originate at a common point called the vertex. One of the rays is called the initial side of the angle and the other ray is called the terminal side. Angles that share the same initial side and terminal side are said to be coterminal. To find a co terminal angle to some angle α (in degrees): Radians the other angle measure. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.

84 AMHS PreCalculus 84 Conversion between degrees and radians 360 = radians. Therefore, a) 1 radian = degrees b) 1 = radians Ex. 1 Convert the following radian measure to degrees: a) 5π 6 b) π 10 c) 4π d) 3 Ex. Convert the following degree measure to radians: a) 400 b) 10 To find a co terminal angle to some angle α (in radians): Ex. 3 Find a coterminal angle, one positive and one negative, to 5 π. 3

85 AMHS PreCalculus 85

86 AMHS PreCalculus 86 We use the unit circle to quickly evaluate the trigonometric functions of the common angle found on it. To summarize how to evaluate the Sine and Cosine of the angles found on the unit circle: 1. sin(α ) =. cos(α )= Ex1. Find the exact value π a) sin( ) 4 π b) cos( ) π c) sin( ) 6 π e) sin(150 ) d) cos( ) 3 f) 11π cos( ) 6 g) 3π sin( ) h) 5π i) sin(330 ) cos( ) 3 Ex. Find the exact value by finding coterminal angles that are on the unit circle. a) 13π sin( ) 4 b) 7π sin( ) 6 c) sin( 300 )

87 AMHS PreCalculus 87 The Cosine is an even function The Sine is an odd function cos( α ) = cos( α ) sin( α ) = sin( α ) Ex. 3 Find the exact value a) 7π cos( ) b) 6 3π sin( ) c) 4 13π cos( ) 4 Reference Angles ' For any angle α in standard position, the reference angle ( α ) associated with α is the acute angle formed by the terminal side of α and the x axis. Ex. 1 Find the reference angle ' α for the given angles. a) α = π 3 b) α =.3 c) 5π α = d) 4 α = 5π 3

88 AMHS PreCalculus 88 The signs (+ or value) of the Sine, Cosine and Tangent functions in the four quadrants of the Euclidean plane can be summarized in this way: Ex. Find the exact value a) π sin( ) 3 b) 5π cos( ) 6 c) 5π sin( ) 3 π d) cos( ) 3 e) cos( 300 ) f) sin(150 ) Ex.3 Find all values of θ in the interval [0, π ] that satisfy the given equation a) sin( θ ) = b) cos( θ ) = 3 Ex. 5 If 3π sin( t ) = and π t, find the value of cos( t). 3

89 AMHS PreCalculus 89 Arc Length In a circle of radius r, the length s of an arc with angle θ radians is: s= rθ Ex. 1 Find the length of an arc of a circle with radius 5 and an angle 5 π. 4 Ex. Find the length of an arc of a circle with radius 13 and an angle30. Ex. 3 The arc of a circle of radius 3 associated with angle θ has length 5. What is the measure of θ? Area of a Circular Sector In a circle of radius r, the area A of a circular sector formed by an angle of θ radians is A = 1 r θ Ex. 1 Find the area A of a sector with angle 45 in a circle of radius 4.

90 AMHS PreCalculus 90 Graphs of the Sine and Cosine Functions Ex. 1 Graph f ( x) = sinx Domain: Range: x intercepts: Period: Amplitude: Even or odd? Ex. Graph f ( x) = cosx Domain: Range: x intercepts: Period: Amplitude: Even or odd?

91 AMHS PreCalculus 91 Ex.3 Graph one period of each function a) f ( x) = cos( x) b) f ( x) = 1+ sin( x) π c) f( x) = cos( x ) d) f ( x) = sin( x)

92 AMHS PreCalculus 9 Graphs of f ( x) = Asin( Bx+ C) + Dand f ( x) = Acos( Bx+ C) + Dwhere A 0 and B > 0 have: Amplitude: Period: Horizontal shift (Phase shift): Vertical shift: Ex.1 Graph one period of each function. π a) f( x) = sin( x ) 3 Amplitude: Period: Horizontal Shift: End Points: b) 1 π f( x) = 3sin( x+ ) 4 Amplitude: Period: Horizontal Shift: End Points:

93 AMHS PreCalculus 93 π c) f( x) = 1+ cos( x ) 4 Amplitude: Period: Horizontal Shift: End Points: Ex. Write a Sine or Cosine function whose graph matches the given curve. a) x scale is 4 π b) x scale is π Ex. 3 Write a Sine and Cosine function whose graph matches the given curve. x scale is 3 π

94 AMHS PreCalculus 94 Unit 7 Trigonometry Part Revisiting Tangent, Cotangent, Secant and Cosecant These are called the Quotient Identities: a) sin( α) tan( α) = b) cos( α) cos( α) cot( α) = sin( α) The following are called the Reciprocal Identities: a) c) e) 1 csc( α) sin( α) b) 1 cot( α) tan( α) d) 1 sin( α) csc( α) f) 1 sec( α) = cos( α) 1 tan( α) = cot( α) 1 cos( α) = sec( α) Ex. 1 Evaluate all six trigonometric functions at the following values of θ : π π a) θ = b) θ = 6

95 AMHS PreCalculus 95 The Pythagorean Identities: a) sin ( x) + cos ( x) = 1 Using the Quotient and Reciprocal identities we can derive the other two Pythagorean Identities: sin ( x) + cos ( x) = 1 sin ( x) + cos ( x) = 1 Conclusion The other two identities are: b) c) Ex. Find the values of all six trigonometric functions from the given information: a) 4 sin( θ ) =, θ is in the first quadrant. b) csc( α ) = 5, 3 π < α < π 5

96 AMHS PreCalculus 96 Graphs of the Tangent and Cotangent Functions Ex. 1 Graph f ( x) = tanx Domain: Range: x intercepts: Period: Even or odd? Ex. Graph f ( x) = cotx Domain: Range: x intercepts: Period: Even or odd?

97 AMHS PreCalculus 97 Ex.3 Sketch one period of each function e) f ( x) = tan( x) π f) f( x) = cot( x+ ) 4 Ex. 4 Find the period of the following functions: f ( x) = tan( x) γ π x f ( γ ) = tan( ) f( x) = cot( ) 3 Ex. 5 Find all the values of t in the interval [0, π ] satisfying the given equation: a) tan( t ) + 1 = 0 b) cot( t ) + 3 = 0

98 AMHS PreCalculus 98 Graphs of the Secant and Cosecant Functions Ex. 1 Graph f ( x) = secx Domain: Range: x intercepts: Period: Even or odd? Ex. Graph f ( x) = cscx Domain: Range: x intercepts: Period: Even or odd?

99 AMHS PreCalculus 99 Ex.3 Graph one period of each function g) 1 π π f( x) = sec( x ) h) f( x) = csc( x ) 4 More on Trigonometric Identities Ex. 1 Use the identities you have learned so far to verify the following: a) 3 sin( θ )cos ( θ) sin( θ) = sin ( θ) b) (1 (cos x) )(sec x) = (tan x)

100 AMHS PreCalculus 100 c) 1+ sinα cos α = (sinα + cos α) d) cot x + tan x = sec xcsc x Sum and Difference formulas for Sine and Cosine sin( α + β) = sinαcos β + cosαsin β cos( α + β) = cosαcos β sinαsin β sin( α β) = sinαcos β cosαsin β cos( α β) = cosαcos β + sinαsin β Ex. Use the sum and difference formulas to determine the value of the following trigonometric functions. a) π 3π sin( + ) 6 4 b) 7π cos( ) 1 Ex. 3 Verify the identity: π sin( t+ ) = cost