5.4 - Quadratic Functions

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1 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Quadratic Functions Definition: A function is one that can be written in the form f (x) = where a, b, and c are real numbers and a 0. (What do we have if a=0? ) This form is called the of a The quadratic form f (x) = a(x h) 2 + k is called The graph of a quadratic function is called a Definition: Extreme functional values are the maximum and/or minimum values of the function. The x-intercepts of a quadratic function are also called the or the. 1. Let g(x) = (x 2) 2 +1 Determine the a) vertex b) axis of symmetry c) maximum value of the function d) x-intercepts

2 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 93 It is nice when a quadratic function is given in standard form because it makes it easy to see that the vertex is at Also determining the x-intercepts is relatively easy. When a quadratic function is given in general form it is easy to determine that the y-intercept is, but determining the vertex is no longer easy. Sometimes you will want to convert a quadratic function from general form into standard form by Observations for understanding completing the square: Expand the following expressions, looking for patterns: (x 3) 2 = x 2 x (x + 4) 2 = x 2 x (x 5) 2 = x 2 x (x 9) 2 = x 2 x Now work backward... x 2 +16x = (x ) 2 x x = (x ) 2 x 2 14x = (x ) 2 x 2 12x = (x ) 2

3 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 94 Converting quadratics from general form to standard form 2 Let f (x) = x 2 2x + 4 Put the function in standard form to determine the a) vertex b) axis of symmetry c) minimum value of the function d) x-intercepts 3. Let g(x) = x 2 8x +17 Put the function in standard form to determine the a) vertex b) axis of symmetry c) minimum value of the function d) x-intercepts

4 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Let h(x) = x 2 5x + 6 Put the function in standard form to determine the a) vertex b) axis of symmetry c) minimum value of the function d) x-intercepts 5. Let f (x) = 3x 2 +12x + 4 Put the function in standard form to determine the a) vertex b) axis of symmetry c) minimum value of the function d) x-intercepts

5 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Let g(x) = 4x 2 +12x + 7 Put the function in standard form to determine the a) vertex b) axis of symmetry c) maximum value of the function d) x-intercepts 7. Consider the function f (x) = ax 2 + bx + c Determine a) the vertex! b) the axis of symmetry c) the extreme value of the function

6 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 97 d) x-intercepts We have just derived When ax 2 + bx + c = 0, x = Notice that axis of symmetry is always half way between the zeros. b 2 4ac is called It is often denoted by an uppercase D = When a quadratic function starts in general form but is converted to standard form, and the x- intercepts are determined from standard form, it is called, solving by completing the square. When a quadratic function starts in general form, but it is not necessary to determine the vertex, the quadratic formula can be used to determine the x-intercepts. Now here is a third technique for determining the x-intercepts of a quadratic function. Finding the Zeros of Quadratic Functions by Factoring Another technique for solving quadratic equations is by factoring. This technique is based on the Zero-Product Principle. Think about it this way: If a, b! and if ab = 1, must either a = 1 or b = 1? If ab = 0, must either a = 0 or b = 0?

7 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 98 To find where the graph of f (x) = x 2 + 7x + 6 intersects the x-axis, we solve x 2 + 7x + 6 = 0. x 2 + 7x + 6 =, So if = 0 then either = 0 or = 0 so or are solutions to the equation x 2 + 7x + 6 = 0 Consider the quadratic function f (x) = x 2 + x 6. f ( 3) = and f (2) = and f (x) can be factored so that f (x) = This is not a coincidence! Consider the quadratic function f (x) = ax 2 + bx + c. f (m) = 0 and f (n) = 0 iff f (x) can be factored so that f (x) = a( x m) ( x n) In other words if a parabola crosses the x -axis at say x = 2 and x = 4, then the factors of the parabola s quadratic function are and If you know that a parabola intersects the x -axis at x = 3 and x = 2, can you be sure that its function is f (x) = (x + 3)(x 2)? It could be that f (x) = or f (x) =

8 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 99 Factoring quadratics, lead coefficient 1: To factor x 2 +10x +16, determine if there are 2 numbers whose product is and whose sum is. Those two numbers are and so x 2 +10x +16 = To factor x 2 2x 48, determine if there are 2 numbers whose product is and whose sum is. Those two numbers are and so x 2 2x 48 = Factoring quadratics with lead coefficient is not 1 Use the Blankety-Blank Method! To factor: 10x 2 +11x + 3 determine if there are 2 numbers whose product is and whose sum is Those two numbers are and Now rewrite 10x 2 +11x + 3 = To factor: 3x 2 14x 5 determine if there are 2 numbers whose product is and whose sum is Those two numbers are and Now rewrite 3x 2 14x 5 = Important Fact: a 2 + b 2 does not factor over the real numbers. So x cannot be factored using real numbers. Notice that the graph of f (x) = x does not intersect the x -axis.

9 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 100 Factor using special patterns: Look what happens when you multiply and simplify: (x 3)(x + 3) = Difference of Squares a 2 b 2 = Square of a Binomial a 2 + 2ab + b 2 = a 2 2ab + b 2 = 8. Find the x-intercepts of the following functions by factoring a) f (x) = x b) g(x) = 16x 2 81 c) h(x) = x 2 +10x + 25 d) p(x) = 16x 2 + 8x +1 e) r(x) = 6x 2 + 7x 5 f) v(x) = 20x 2 + 7x 3 9. Find the x-intercepts of the following functions using the quadratic equation a) f (x) = x b) g(x) = 16x 2 81

10 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Given the parabola on the right, determine the quadratic function that yields this graph. Then find the determinant of that quadratic. When the discriminant is there are real roots. In these cases, the parabola intersects the x -axis.

11 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Given the parabola on the left, determine the quadratic function that yields this graph. Then find the determinant of that quadratic. When the discriminant is there is real root. It is sometimes called a. In these cases, the parabola intersects the x -axis.

12 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Given the parabola on the left, determine the quadratic function that yields this graph. Then find the determinant of that quadratic. When the discriminant is there are real roots. In these cases, the parabola intersect the x -axis.

13 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 104 Applications 13. A toy rocket is launched from a 304 foot cliff. If the height, in feet, of the rocket is given by.h(t) = 16t t where t is the number of seconds after liftoff. Notice h(0) = This corresponds to the rocket starting What kind of function is h(t)? So the graph of this function is a The maximum value of this function will occur at the Rewrite h(t) in standard form: a) find the maximum height of the rocket. b) How long does it take for the rocket to reach that maximum height? c) When does the rocket hit the ground? d) What is the domain of h(t)?

14 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! Joe has 30 ft of fence to make a rectangular kennel for his dogs, but plans to use his garage as one side. What dimensions produce the greatest area? Definition: Extreme functional values are the maximum and/or minimum values of the function on its domain. Minimum value Minimum value Minimum value Maximum value Maximum value Maximum value

15 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 106 Local Extrema There is a local maximum at x = a if there is an open interval I containing a, and x I It is also important to notice on the graph that at a local maximum, the graph changes from There is a local minimum at x = b if there is an open interval I containing b, and x I It is also important to notice on the graph that at a local minimum, the graph changes from 15. Consider the graph of the function f (x) = x 3 x 2 6x. a) What is the approximate value for the local maximum? b) About where does this local maximum occur? c) Find an approximate value for the local minimum. d) Indicate the approximate location of this minimum value. Note: Sometimes you will want to know the maximum (or minimum) value of the function.! Sometimes you will want to know where (for what x value) the maximum value occurs.

16 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 107 Extra Problems for Quadratic Functions 1. The maximum value of the function g(x) = (5 x)(x + 3) 6 is 2. Here is a quadratic function in general form: f (x) = 3x x + 28 Convert it to standard form: 3. Determine the coordinates of the x-intercepts of the graphs of a) f (x) = 5x 2 7x + 2 b) g(x) = 4x x + 49 c) h(x) = 4x x + 36

17 Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! As part of a holiday tradition physics students went to the roof of the building and fired an orange projectile straight up into the air. They were able to determine that the height in feet of the projectile could be described by the function f (t) = 16t t + 96 where t is time in seconds after the projectile was fired into the air. a) How tall was the building? ft b) How high did the projectile go? ft c) How long until it smashed onto the ground? sec 5. If ax 2 + bx + c = 0,!!! then x =

18 Fry TAMU Spring 2017 Math 150 Notes Section 2.2 Page Factoring (Polynomials) Which of these are examples of polynomials? Polynomial Not a polynomial Polynomials should not have x 2 + 2x +1 x 7 + π ( 2) x 2 + 2x +1 4 x 1 1 x + 3x2 ( 5x 3 + 3x 2 2) 1 4 ( 5x 3 + 3x 2 2) 2 x 2 + sin x 5 Polynomials are expressions of the form a n x n + a n 1 x n 1 +!+ a 1 x + a 0 where a i! and n! n is called a n is called a 0 is called If n = 2, the polynomial is called a Example: If n = 3, the polynomial is called a Example: A polynomial with 2 terms is called a Example: A polynomial with 3 terms is called a Example:

19 Fry TAMU Spring 2017 Math 150 Notes Section 2.2 Page 110 Factoring polynomials helps us when we need to find the zeros of polynomials. Techniques for Factoring Polynomials Common Factors: 12x 4 + 4x 2 + 2x = Factor by Grouping: 3x 3 + x 2 12x 4 (Factoring by grouping is your best bet if you have a long cubic!) 3x 3 + x 2 12x 4 = Look what happens when you multiply (a + b)(a 2 ab + b 2 ) Similarly look what happens when you multiply (a b)(a 2 + ab + b 2 ) Sum of Cubes a 3 + b 3 Difference of Cubes a 3 b 3

20 Fry TAMU Spring 2017 Math 150 Notes Section 2.2 Page Factor a) 27 x 3 b) 64x 3 +1 c) x6 + 8 d) x 5 4x 3 x e) 3x3 + 6x 2 12x 24 f) 16x 4 25x 2 Factoring polynomials in quadratic form 2. Factor a) x 4 5x 2 6 b) x 6 + 2x 3 +1

21 Fry TAMU Spring 2017 Math 150 Notes Section 2.2 Page 112 c) 4x 12 9x Factoring some non-polynomials with similar techniques 3. Factor 2 3 a) x + 3 x + 2 b) 12x x c) x 2 + 3x + 4x 1 3

22 Fry TAMU Spring 2017 Math 150 Notes Section 2.2 Page 113 Extra Problems for Factoring 1. Factor completely over the integers a) 27x 4 64x = b) 8ax 6x 12a + 9 = = c) 169a 6 121b 4 = d) 6x 2 +13x 5 = 2. Factor completely over the real numbers x 2 2 = 3. Factor completely x =

23 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 114 Chapter 6.1 (Graphing) Polynomial Functions Polynomial Degree Leading Term Constant Term f (x) = 4x 3 + 2x 5 g(x) = 17x 5 + 6x 3 h(x) = πx 6 17 c(x) = 14 p(x) = p n x n + p n 1 x n 1 +!+ p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers, the p i 's are real numbers and p n 0 r(x) = 2 + 3x 5x 4 When we graph polynomials, we ll pay special attention to! the x and y intercepts! where the graph is above or below the x-axis! the behavior of the function as x approaches.! the behavior of the function as x approaches. We will use notation like this: As x, x 3. It reads, As x goes to infinity, x cubed goes to infinity. In this case it means that when the values of x are very large and positive, x cubed is very large and positive. Notice how the graph of the function goes up on the right side. As x, x 3. This reads, As x goes to negative infinity, x cubed goes to negative infinity. It means that when the values of x are very large and negative, x cubed is very large and negative. In this case, the graph of the function goes down on the left side. Later we will study functions that level off for large values of x and other functions that oscillate forever. In calculus you will learn a precise definition for very large.

24 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 115 Cubics: f (x) = x 3!!!!!!!! g(x) = (x 2) 3 Constant Term: y-intercept: Leading Term: Constant Term: y-intercept: Leading Term: As x, x 3 As x, (x 2) 3 As x, x 3 As x, (x 2) 3 x-intercept: Notice that the graph only intersects the x- axis in one place -- at x =. This is the only place that the function changes sign. When x < 0, x 3 < 0 When x > 0 x 3 > 0 x-intercept: Notice that the graph only intersects the x- axis in one place -- at x =. This is the only place that the function changes sign. When x < 2, (x 2) 3 > 0 When x > 2 (x 2) 3 < 0

25 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 116 h(x) = 2(x + 3)(x 1) 2 Constant Term: So the y-intercept is Leading Term: As x, 2(x + 3)(x 1) 2 As x, 2(x + 3)(x 1) 2 x-intercepts: Now we would like to solve 2(x + 3)(x 1) 2 > 0 (because when 2(x + 3)(x 1) 2 > 0, the graph of h(x) will be the x-axis.) Redraw the x-axis and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product h(x) > 0 when so that is where the graph of h(x) is above the x-axis

26 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 117 p(x) = 1 4 (x + 2)(x 2)(3x 4) Constant Term: So the y-intercept is Leading Term: As x, 1 4 (x + 2)(x 2)(3x 4) As x, 1 4 (x + 2)(x 2)(3x 4) x-intercepts: Solve p(x) > 0. Draw a number line and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product p(x) > 0 when so that is where the graph of p(x) is above the x-axis

27 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 118 True or False: Some cubics never intersect the x-axis. Some cubics intersect the x-axis in exactly one place. Some cubics intersect the x-axis in exactly two places. Some cubics intersect the x-axis in three places. Some cubics intersect the x-axis in four places. Quartics f (x) = x 4 +1!!!!!!! g(x) = 1 4 (x 2) 4 Constant Term: y-intercepts: x-intercepts: Leading Term: As x, x 4 +1 As x, x 4 +1 Constant Term: y-intercepts: x-intercepts: Leading Term: As x, 1 (x 2) 4 4 As x, 1 (x 2) 4 4

28 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 119 h(x) = x 3 (2x + 7) y-intercept Leading Term: As x, x 3 (2x + 7) As x, x 3 (2x + 7) x-intercepts: Solve h(x) = x 3 (2x + 7) > 0 p(x) = (x 2 1)(x + 2) 2 y-intercept Leading Term: As x, p(x) = (x 2 1)(x + 2) 2 As x, p(x) = (x 2 1)(x + 2) 2 x-intercepts: Solve p(x) = (x 2 1)(x + 2) 2 > 0

29 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 120 Polynomials of Higher Degree In general, to understand the behavior of a polynomial, 1. Plot the y-intercept 2. Determine the behavior of the polynomial for large positive values of x and for large negative values of x. The behavior of the polynomial at these extremes will be dominated by the leading term, the term with the highest power of x. 3. Find the x-intercepts 4. Determine where the polynomial is positive and negative because this will tell you where the graph is above and below the x-axis. f (x) = x 5!!!!!!! g(x) = x 5!!! Constant Term: Leading Term: Constant Term: Leading Term: As x, x 5 As x, x 5 As x, x 5 As x, x 5 x-intercepts: x-intercepts: x 5 > 0 x 5 > 0

30 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 121 h(x) = (3x 1) 2 (x 2)(x +1)(x + 3)!!!!!!! y-intercept is Leading Term: As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) x-intercepts: Solve h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) > 0

31 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 122 f (x) = x(x 3) 2 (2x + 5) 2 Degree of the polynomial: y-intercept: Leading Term: As x, x(x 3) 2 (2x + 5) 2 As x, x(x 3) 2 (2x + 5) 2 x-intercepts: Solve: x(x 3) 2 (2x + 5) 2 > 0

32 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 123 True or False: All linear functions intersect the x-axis. True or False: All polynomial functions of degree zero intersect the x-axis. True or False: All polynomial functions of degree one intersect the x-axis. True or False: All quadratic functions intersect the x-axis. True or False: All cubic functions intersect the x-axis. True or False: All quartic functions intersect the x-axis. True or False: All quintic functions intersect the x-axis. True or False: All polynomials functions of even degree intersect the x-axis. True or False: All polynomials functions of an odd degree intersect the x-axis.

33 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page 124 Extra Problems for Section a) If f (x) = 7x 5 + 6, then as x, f (x) b) If f (x) = 7x 5 + 6, then as x, f (x) c) If f (x) = 7x 4 + 6, then as x, f (x) d) If f (x) = 7x 4 + 6, then as, x, f (x) 2 a) f (x) = x( x 1) 2 b) g(x) = x( x 1) 2 c) h(x) = x( x +1) 2 d) p(x) = x( x +1) 2 e) r(x) = x( x +1)(x 1) 3 a) f (x) = x( x +1) ( x 1) b) g(x) = x( x +1) ( x 1) c) h(x) = x 2 ( x +1) 2 ( x 1) d) p(x) = x 2 ( x +1) 2 ( x 1) e) r(x) = x 2 ( x +1) ( x 1) 2

34 Fry TAMU Spring 2017 Math 150 Notes Section 6.1 Page a) f (x) = x( x +1) 2 ( x 1) b) g(x) = x( x +1) 2 ( x 1) c) h(x) = x( x +1) ( x 1) 2 d) p(x) = x( x +1) ( x 1) 2 e) r(x) = ( x +1) 2 ( x 1) 2 5 a) f (x) = ( x +1) 2 ( x 1) 3 b) g(x) = ( x +1) 2 ( x 1) 3 c) h(x) = ( x +1) 3 ( x 1) 2 d) p(x) = ( x +1) 3 ( x 1) 2 e) r(x) = x( x +1) ( x 1) 6 a) f (x) = ( x +1) ( x 1) 3 b) g(x) = ( x +1) ( x 1) 3 c) h(x) = ( x +1) 2 ( x 1) 3 d) p(x) = ( x +1) 2 ( x 1) 3 e) r(x) = ( x +1) 2 ( x 1) 2

35 Fry TAMU Spring 2017 Math 150 Notes Section 2.1 Page! (Dividing) Polynomials Dividing Polynomials (This is a lot like long division of integers!) 1. 4x 3 x 2 5x + 2 x 1 = What is the dividend? What is the divisor? What is the quotient?

36 Fry TAMU Spring 2017 Math 150 Notes Section 2.1 Page! x 3 2x x 2 1 =! What is the dividend? What is the divisor? What is the quotient?

37 Fry TAMU Spring 2017 Math 150 Notes Section 2.1 Page! Use polynomial long division to determine 16x 4 + 8x 3 + 2x +1 = 4x 2 2x +1

38 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page! Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values where the denominator is 1. g(x) = 1 x +1 a) domain b) y -intercept c) x -intercept(s) A fraction is zero when its is zero and its is NOT zero. That is why the graph of 1 x +1 never the. (The numerator is never zero.) d) vertical asymptote Vertical asymptotes occur where the is zero, but the is not zero. e) critical value(s) for g(x) f) Solve g(x) > 0 g) for large x, g(x) acts like, (This is the quotient of the.) so as x, g(x) --> and as x g(x) --> When the value of a function approaches a constant value for large values of x, we say that the graph of the function has a The graph of g(x) has a horizontal asymptote of

39 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page! h(x) = x x +1 a) domain b) y -intercept! c) x -intercept(s) d) vertical asymptote e) critical value(s) for h(x) (Critical values are those that make either the numerator or the denominator zero.) f) Solve h(x) > 0 g) for large x, (think about estimating) h(x) acts like, (This is the quotient of the ) so as x, h(x) --> and as x h(x) --> Here we say that h(x) has a horizontal asymptote of

40 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page! p(x) = x2 +1 x +1 a) domain b) y -intercept! c) x -intercept(s) d) vertical asymptote e) critical value(s) for p(x) f) Solve p(x) > 0 g) For large x, (think about estimating) p(x) acts like, (This is the quotient of the ) so as x, p(x) --> and as x p(x) --> Does the graph of p(x) have a horizontal asymptote? Some people would say that the graph of p(x) has a asymptote.

41 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page!132 A hole in the graph indicates a place where the function is undefined, but the function s behavior is not asymptotic. Understanding Holes in Graphs: 4. f (x) = x2 1! a) domain x +1 b) y -intercept! c) x -intercept(s) d) vertical asymptote(s) e) there is a hole at f) the y - coordinate of hole g) critical value(s) for f (x) h) Solve f (x) > 0 i) For large x, f (x) acts like, so as x, f (x) --> and as x f (x) --> j) horizontal asymptote

42 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page! g(x) = x2 1 x 2 +1! a) domain b) y -intercept! c) x -intercept(s) d) vertical asymptote(s) e) there is a hole at f) the y - coordinate of hole g) critical value(s) for g(x) h) Solve g(x) > 0 i) For large x, g(x) acts like, so as x, g(x) --> and as x g(x) --> j) horizontal asymptote

43 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page!134 x h(x) = 3( x 2 9)(x 1)! a) domain b) y -intercept! c) x -intercept(s) d) vertical asymptote(s) e) there is a hole at f) the y - coordinate of hole g) critical value(s) for h(x) h) Solve h(x) > 0 i) for large x, h(x) acts like, so as x, h(x) --> and as x h(x) --> j) horizontal asymptote

44 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page! p(x) = 2x 3 + 6x 2 x 3 + 3x 2 4x 12 a) domain b) y -intercept! c) x -intercept(s) d) vertical asymptote(s) e) there is a hole at f) the y - coordinate of hole g) critical value(s) for p(x) h) Solve p(x) > 0 i) For large x, p(x) acts like, so as x, p(x) --> and as x p(x) --> j) horizontal asymptote

45 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page!136 To graph a rational function, a) Evaluate the function at x = 0, this is the b) Find the values for x for which the numerator is zero, but the denominator is not zero. This is where the graph c) Find the values for x for which the denominator is zero, but the numerator is not zero. This is where the graph d) Find the values of x for which both the numerator and the denominator are zero. This is where there is e) To find the y - coordinate of the hole: If there is a hole at x = a, then ( x a) is a factor of both the numerator and the denominator. The rational function f (x) can be written in the form (x a) f (x) = p(x) ( x a) q(x). It could be that p(x) = 1 and/or q(x) = 1.! Let f (x) = p(x) q(x), then the y - coordinate of the hole is f a ( ). f) Simplify the quotient of the leading terms of the numerator and the denominator. The end behavior of this function is the same as the end behavior of the given function. g) If the function tends to a constant c as x gets very large, then we say that the graph has a horizontal asymptote of y = c. h) Determine where the function is greater than 0.! This is where the graph of the function is.

46 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page!137 Extra Problems for Section f (x) = x2 x 12 2(x 2 16) a) List the coordinates of all of the x -intercepts.! If there are none, write NONE in the blank provided. b) List the equations of all of the vertical asymptotes.! If there are none, write NONE in the blank provided. c) List the equations of all of the horizontal asymptotes.! If there are none, write NONE in the blank provided. d) List the coordinates of all the holes.! If there are none, write NONE in the blank provided. e) On what intervals is the graph of f (x) = x2 x 12 2(x 2 16) Write your answer using interval notation: above the x-axis?

47 Fry TAMU Spring 2017 Math 150 Notes Section 6.2 Page! g(x) = 8x5 3 7x 4 1!! as x, g(x) --> 3. h(x) = 8x4 3 7x 5 1!! as x, h(x) --> 4. f (x) = x2 + 5x + 6 x 2 + 2x State the domain of f (x) b) List the coordinates of all of the x -intercepts. c) List the coordinates of all of the y -intercepts. d) List the equations of all of the vertical asymptotes. e) List the coordinates of all the holes. f) List the equations of all of the horizontal asymptotes

48 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! 139 Bonus Section: Rational Expressions (This is not in your text book.) A rational expression is the quotient of two In this section we ll add, subtract, multiply, and divide rational expressions. Our goal will be to simplify the result so that it is expressed as the quotient of two factored polynomials. We will also need to be aware of possible restrictions on the values of our independent variable(s), usually x. Rational expressions are undefined when So the rational expression x = as long as x 1. For the following rational expressions, list any restrictions that exist for x. a) x 1 x!!!!!! Restrictions on x : b) x + 2!!!!!! Restrictions on x : x c) x7 x 4!!!!!! Restrictions on x : x 5 3 x What is the difference between a rational expression and a rational function? Very little. x +1 x + 2 is a rational expression whereas f (x) = x +1 x + 2 is a rational function. Simplifying rational expressions is like reducing rational numbers: = = 5 6 Notice that = 3+12, but we would not cancel the 12 s! When simplifying rational expressions we look for common in the numerator and denominator.

49 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! 140 In the previous section, the rational functions were already factored. Sometimes they are found in a messier form. In this section we ll practice the algebraic skills necessary to write a given rational function as the quotient of two factored polynomials. 2. f (x) = 2 x x 3 Domain

50 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! g(x) = x 4 x +1 x2 8x +16 x 2 1 Domain 4. h(x) = 3 x + 2 i x 2 x 2 x 2 1 Domain

51 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! f (x) = x2 x 6 (x 2 + 7x +12) x + 4 Domain 6. g(x) = x 3 1 x +1 x 1!!! Domain x 2 + 2x +1

52 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! 143 A compound or complex fraction is an expression containing fractions within the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division. 7. h(x) = 3 x + 1 2x 7 5 x +1 Begin by getting a common denominator in the numerator and a common denominator in the denominator.

53 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! 144 This is a function of 2 variables. We won t try to graph it, we ll just simplify it. 8. f (x, h) = 3 ( x + h) 3 2 x 2 h Note: In the traditional presentation of Math 150, students learn to simplify rational expressions before the concepts of function and domain are defined. So in some WebAssign problems for 1D, you ll be asked for the restrictions on x. You will want to report the values of x that make the function undefined even before it is simplified.

54 Fry TAMU Spring 2017 Math 150 Notes Bonus Section I Rational Expressions Page! 145 Extra Problems for Bonus Section I Rational Expressions: 1. Fully simplify: a) 3x 1 + 7x x 1 7x 2 b) 3 x 2 3 x 3 x 2 c) 9 1 x x + 1 x 2

55 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page! Trigonometric Functions (the second part) Another way to define the trigonometric functions: Given a circle of radius one, (often called the unit circle) with a radial line drawn at an angle θ, measured counterclockwise from the positive x-axis, the radial line intersects the circle at a point (x, y). The trigonometric functions can then be defined as sinθ cosθ tanθ cscθ secθ cotθ Find cos 3π 4

56 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page!147 Let θ be an angle in standard position. The reference angle θ R is the acute angle formed by the terminal side of θ and the x -axis. cos 5π 6 sin 4π 3

57 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page!148 sin π 6 tan 2π 3 Reference Angle Theorem: Let trig( θ ) be any one of the six trigonometric functions defined above (on page 146).!!!! Then trig( θ ) = ±trig( θ R )!!!! The correct sign is determined by the quadrant of θ.

58 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page!149 tan 7π 6 csc 5π 3

59 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page! Given that sinθ = 3 5 and θ is acute, determine the values of a) cosθ = b) tanθ = 2. Given that sinθ = 3 5 and θ is NOT acute, determine the values of a) cosθ = b) tanθ =

60 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page! Given that tanθ = 2 and cosθ < 0 determine the value of sinθ. 4. Given that secθ = 7 5 and tanθ < 0 determine the value of tanθ.

61 Fry TAMU Spring 2017 Math 150 Notes Section 9.2 Page!152 Extra Problems for Section 9.2 If tanθ = 6 5 and cosθ < 0, then sinθ = From pages 737 and 738

62 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! Solving Equations The solution of an equation is a value (or a set of values) that yields a true statement (an identity) when substituted for the variable in an equation. The word solve means determine Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2x +1 = 4 and 4x + 2 = x 3 The algebraic techniques to find the solutions to these equations are simple, but I want you to keep in mind that there is a geometric interpretation associated with the equation. We are looking for the intersection of the graphs of two linear functions. The solutions we find are the x-coordinates of the intersections of the graphs. Here are the graphs of!!!!!! Here are the graphs of and!!!!!! and!!!!!!! See how the lines intersect at!!!!! See how the lines intersect at x =!!!!!!! x = See how substituting x =!!!! See how substituting x = into 2x +1 = 4!!!!!!!! into 4x + 2 = x 3 makes the statement true.!!!!!! makes the statement true.

63 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 154 Solving Quadratic Equations Quadratic equations are equations involving only polynomials of degree two. Examples include x 2 x 6 = 2x 2! and! 5x 2 5x = 2x 2 3x +1 Geometrically, such equations could represent the intersection of a parabola and a line or the intersection of two parabolas. The solutions we find are the x-coordinates of the intersections of the graphs. x 2 x 6 = 2x 2 5x 2 5x = 2x 2 3x x 2 = x 1

64 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 155 Solve Quadratic Equations by Factoring 1. Solve Put in general form a) 1 2x 2 = x Factor Set each factor equal to zero Put in general form b) 6x 2 = x +15 Factor Set each factor equal to zero

65 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 156 Solve by completing the square. 2. Solve the following quadratic equation by completing the square: 2x 2 + x 8 = 0 If ax 2 + bx + c = 0, then x = This is called 3. Solve using the quadratic formula: 2x 2 = 6x 3

66 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 157 Solving equations in quadratic form 4. Solve a) x 4 5x 2 6 = 0 b) x x 5 +1 = 0 c) 4x 12 9x = 0

67 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 158 d) x x = 0

68 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 159 Solving Rational Equations Rational equations are equations involving rational functions. Solving rational equations is equivalent to finding the intersection(s) between the graphs of 2 rational functions. Algebraically what we do is to set one side of the equation equal to zero and remember that a fraction is zero when The mistake: Consider solving 2 x = 3 x. A lot of people want to start by cross multiplying or multiplying both sides of the equation by x. This would yield. But neither 2 nor 3 are defined. This equation does not have a solution. The risk occurs when multiplying both sides of an equation by something that could be zero. 5. Solve a) 1 2x = x The algebra will give us the x-coordinate of the intersection.

69 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 160 b) 4 x 4 3 x 1 = 1

70 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 161 Radical Equations To Solve Radical equations: I. Isolate one radical. II. Raise both sides of the equation to the appropriate power to remove the radical.! (Usually this means square both sides of the equation.) III. Repeat the process until all radicals have been removed IV. Check for extraneous solutions! Why is it important to check for extraneous solutions to equations involving radicals? How many solutions exist for the equation x = 3? Square both sides of the equation: What are the solutions to this equation? See how is not a solution to the original equation?! REMEMBER: (a + b) 2 a 2 + b 2 (a + b) 2 = 6. Solve a) 3 = x + 2x 3

71 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 162 b) 8x +17 2x + 8 = 3 c) x + x 5 = 1

72 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 163 Equations with Absolute Values!! Recall the definition of x = 7. Solve a) x = 4 b) x 3 = 2 c) 8 x + 2 = 7

73 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 164 d) x = 2 e) x 2 x 12 = 8!!

74 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 165 Equations with several variables Equations in science and engineering often include many variables, and it is useful to be able to solve for one of the variables in terms of the others.!! Warm up: If 1 x = , then what is the value of x? 3 8. The following equation comes from the physics of circuits 1 R eq = 1 R R R 3 Solve for R eq.

75 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! T s = 2π m k Solve for k. 10. Solve for y x = 3y 2y 5

76 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! 167 Extra Problems for Section Determine all real solutions of x + x 4 = 4 2. Solve x = 10 9y 3y 5 for y. 3. From Dr. Lynch Fall 2016 Exam 1: Solve the equation A = 2π(tr + s) for r.

77 Fry TAMU Spring 2017 Math 150 Notes Section 3.1 Page! Determine all solutions of a) x 4 + x 3 30x 2 = 0 b) 18x 3 + 9x 2 2x 3 = 0 c) 6x2 +13x + 2 x = 0 d) x 4 + 6x 2 27 = 0

78 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! Solving Inequalities True or False: 4 < 4!!! True or False: 4 4 Also though 3 < 4, -4 < -3 which could also be written -3 > -4. Written more generally, if a < b, then This leads to the rule that when you multiply or divide both sides of an inequality by a negative number, you change the direction of the inequality. 1. Solve a) x < 4 b) x 2

79 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! 170 c) x + 2 < 1 d) 2x 4 3 e) 5 2x > 4

80 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! 171 f) 4 x 3 > 1 Nonlinear Inequalities Consider a,b! ab > 0, then either or If ab < 0 then either or To solve nonlinear inequalities: I. Move every term to one side (make one side zero). II. Factor the nonzero side. III. Find the critical values. (Critical values make the expression zero or undefined.) IV. Let the critical numbers divide the number line into intervals. V. Determine the sign of each factor in each interval. VI. Use the sign of each factor to determine the sign of the entire product or quotient.

81 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! Solve a) x 2 x 6 TRUE or FALSE: 2 < 3 TRUE or FALSE: 2x < 3x because when x < 0, So don t multiply both sides of an inequality by a variable that could be either positive or negative. b) 2 x < 1

82 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! 173 c) 2x + 3 6x +1 > x x Consider the graph of f (x) = 2x + 3 x + 3 It has a VA of the quotient of the leading terms is so the HA is the intercepts are Consider the graph of g(x) = It has a VA of 6x +1 1 x the quotient of the leading terms is so the HA is the intercepts are

83 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! 174 Linear Inequalities 3. Solve a) 1 x > x + 3 b) 1 2x < 17 4x 8 x

84 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! 175 c) 2x + 4 < x + 2 < x d) 4x x 3 x

85 Fry TAMU Spring 2017 Math 150 Notes Section 3.2 Page! 176 Extra Problems for Section 3.2 Solve the inequalities and express your answer in interval notation a) 5 x b) 3x (From Dr. Lynch s Fall 2016 Exam 1.) c) 6 5 x < 2 d) 5 + 2x 3 (From Dr. Lynch s Fall 2016 Exam 1.) x 2

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