Math 2412 Activity 2(Due by EOC Feb. 27) Find the quadratic function that satisfies the given conditions. Show your work!

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1 Math 4 Activity (Due by EOC Feb 7) Find the quadratic function that satisfies the given conditions Show your work! The graph has a verte at 5, and it passes through the point, 0 7 The graph passes through the points 0,0,,0, and,9 The graph passes through the points 0,,,0, and, Sketch the graph of the following polynomial functions 4 4 f 48 5 f 6 f

2 7 a) Use the Intermediate Value Theorem to show that the polynomial function 4 f 4 0, has at least one zero in the interval b) Use the Bisection Method to approimate a zero in the interval 0, Complete the table: Left endpoint (sign) Guess(midpoint) (sign) Right endpoint (sign) Error Bound 0(-) ½ (-) (+) ½ ½ (-) (+) ¼ 4 8 Find the largest area of a rectangle inscribed under the graph of the function the first quadrant by doing the following: a) Epress the area of the rectangle as a function of f in b) What is the domain of the area function? c) Find the maimum value of the area function Find the verte, and determine if it s a maimum or minimum 9 f 0 4 f 8 Graph the following quadratic functions f 5 f 5 4 f

3 4 f 4 Write an equation for the quadratic function whose graph satisfies the conditions 5 Verte:, 4 and passes through,4 6 Passes through the points 0,,,, and,5 7 The -intercepts are and, and the y-intercept is 4 8 The -intercepts are and, and the maimum value is 6 9 A rain gutter is made from sheets of aluminum that are inches wide by turning up the edges to form right angles Determine the depth of the gutter that will maimize its crosssectional area and allow the greatest amount of water to flow 0 A rancher has 000 feet of fence to construct si corrals, as shown in the figure Find the dimensions that maimize the enclosed area y

4 There is an alternative way to multiply two polynomials called the method of detached coefficients It is very similar to the way we multiply numbers As an eample, let s find the product of the polynomials 4 and using this method: From the bottom row, we see that the product is a) Multiply the polynomials detached coefficients 4 7 and 6 using the method of b) Evaluate each of the two polynomials and their product at 0 c) Compare the by-hand multiplication of 47 and 6 to the detached multiplication table in part a) Determine the end behavior of the following polynomials 4 f 4 f 6 The graph to The graph to The graph to The graph to the left goes the right goes the left goes the right goes 4 f 5 f 5 The graph to The graph to The graph to The graph to the left goes the right goes the left goes the right goes

5 Sketch the graphs of the following polynomial functions Label the zeros and the y-intercept 5 5 f 4 6 f 7 8 f f f 4 f f 4 5 f Find a polynomial function of smallest degree with leading coefficient of that satisfies the given conditions 4 Its only zeros are 4,0, The graph crosses the -ais at 4,0, ; lies above the -ais between 4 and 0; lies below the -ais between 0 and 5 Its only zeros are 0 and The graph touches the -ais at 0 and crosses the -ais at ; lies above the -ais between 0 and Divide using long division Divide using synthetic division

6 Use synthetic division to find the indicated function value 44 f 7 5; f 4 45 f ; f 46 Use synthetic division to divide the zeros of f f by Use the result to find all 47 Solve the equation given that is a solution 48 Find k so that is a factor of k 49 Find k so that 4 is a factor of 0 0 k 50 When 7 9 is divided by a polynomial the quotient is and the remainder is Find the polynomial Sketch the graphs of the following rational functions Indicate intercepts and horizontal and vertical asymptotes 5 f 5 f 4 5 f f 4 Sketch the graphs of the following rational functions Indicate intercepts and oblique and vertical asymptotes 4 55 f 56 f 9

7 Find a rational function f p q, in which the degrees of p and q are as small as possible, that satisfies the given conditions(57-58) 57 has a vertical asymptote of, a horizontal asymptote of y 0, y-intercept of, and no -intercept 58 has a vertical asymptote of, a slant asymptote of y, y-intercept of, and - intercepts of and

8 Solve the following inequalities by first completing the sign charts (59-70)

69 0 70 Inequalities of the form something something else get something something else inequalities: (7 and 7) something something else, which simplifies into can be solved by squaring both sides to

10 Inequalities of the form something something else get something something else inequalities: (7 and 7) something something else, which simplifies into can be solved by squaring both sides to Use this idea to solve the following absolute value Find the smallest value of the positive constant m that will make m positive values of {Hint: Consider that m } m 0 for all n n 75 If p a a a a is a polynomial and it has at least n zeros, where n n 0 n 0, then what must be the values of a,, 0 a n? Remember, a polynomial of degree n has at most n zeros(counting repeated zeros)

11 76 If p and q are polynomials of degree n, where 0 p and values, then what can you conclude about 77 Determine all polynomials P so that P P values of P P P P P P problem What if n, and p q for n q? Consider the previous problem and P0 0 Find the,, 5, 6, 6, 6, and use the previous P0? 78 One of the two lines that pass through,0 and are tangent to the parabola ais Find the other line y is the - 79 A fly is crawling from left to right along the top of the parabola y7 A spider waits at the point 4,0 Find the distance between them when they first see each other 80 Find the values of k so that f 4 k has a zero between and

12 8 Find values of a and b so that the polynomial a b For all positive real numbers a and b, show that ab a b ab a b 8 If 0, show that is divisible by {Hint: Multiplying both sides by, leads to Now manipulate it} 84 If, 0 4 y y, show that y {Hint: Multiplying out the left side leads to y 85 If,, 0 9 y z y z, show that y z 86 If,,, 0 6 y z w y z w, show that y z w y y y y } y n 87 What about n for,,, n 0? 88 Suppose that a rational function has vertical asymptotes of and only, has zeros of and 4 only, and has a horizontal asymptote of y a) Sketch the graph of such a function b) Find a formula for such a function 89 Suppose that a rational function has a vertical asymptote of only and a slant asymptote of y a) Sketch the graph of such a function b) Find a formula for such a function

13 Find partial fraction decompositions of the following If 96 If c 0 for all real, then what must be true about c? b 7 0 for all real, then what must be true about b? 97 If a for all real, then what must be true about a? 98 Show that 99 Show that 00 Show that Show that Show that if and y are positive numbers, then y y 0 Show that if 0 y, then y y 04 Find an integer m so that 5 m is an integer

14 05 Find an integer m so that 5 m is an integer 06 For a, b, c, and d positive numbers with a c, show that a a c c b d b b d d Use this result to find 5 different fractions between and 07 f 8 5 Find the domain of the following functions: 08 f 6 Find the domain and range of the following functions: 09 f 0 f f Consider the rational function f a) Find the partial fraction decomposition of f b) Use the partial fraction decomposition from part a) to find the eact value of the sum 4,000,000,000,00 Consider the rational function f a) Find the partial fraction decomposition of f

15 b) Use the partial fraction decomposition from part a) to find the eact value of the sum 4 45,000,000,000,00,000,00 4 Consider the function f 999,998 a) Epress the function f as a rational function b) Use the result of part a) to find the eact value of 4,000,000 5 Is it possible to have? If so, for what values of a and b? a b a b 6 Find a cubic polynomial function p so that p p 7 By eamining the following graph of the functions,,,,, find all values of 0for which a) b)

16 8 Solve the inequality 9 Given that 0 c c c, simplify c 0 Given that a 0 and b 0, simplify a b ab ab Epress as a single radical Find the smallest integer, n, so that n 99 n {Hint: What do you get when you square it?} Use polynomial division to find all the integer values of n so that integer value {Hint: Do some squaring} n n n 4 Find a, b, and c for f a b c if f 4i 0, f 4i 0, and f 0 has an 5 There is an imaginary version of the Rational Root Theorem for finding roots of polynomials: n n If P a a a a is an n th degree polynomial with integer coefficients, and of a n and p n n 0 i is a rational imaginary zero of P, then p r q r q must be a factor of a 0 a) For 4 P 8 6 5, let s look at the possibilities: r must be a factor Possible Rational Imaginary Zero Possible Factor i 5 i 5 Test the possible rational imaginary zeros, and then find all the zeros

17 4 b) For P 4, let s look at the possibilities: Possible Rational Imaginary Zero Possible Factor i i i i 4 Test the possible rational imaginary zeros, and then find all the zeros 6 There is a cubic formula similar to, but more complicated than the quadratic formula Let s derive it in the special case of cubic equations of the form make the substitution u v to get p q u v p u v q 0 u u v uv v pu pv q 0 u v uv p u v q 0 We can solve the cubic if we can find u and v that satisfy This system is equivalent to equation in u q q p u v q p uv p 6 p 0 First, u q u qu 0 This is a quadratic u 7 u, so from the quadratic formula, we get To get v, use the fact that 4 7 So solutions of p q 0 are given by u p v, to get v u 4 p q q 7 or p q q p 4 7

18 q q p p Since the cube roots can produce as many as 4 7 q q p 4 7 numbers, it looks like you could get a lot of solutions, but you will get at most Since the equation is a cubic, once you get one solution, the equation can be reduced to a quadratic a) Use the formula to find one solution of the cubic equation b) Use the formula to find one solution of the cubic equation

f ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation.

f ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation. Test Review Section.. Given the following function: f ( ) = + 5 - Determine the implied domain of the given function. Epress your answer in interval notation.. Find the verte of the following quadratic