# Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama

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2 . The number of values satisfying the equation x 0x x x = x 3 is: Solution: To solve this rational equation, multiply both sides by x x and then solve the resulting cubic equation. x 0x = (x 3(x x x 0x = x 3 8x + x x 3 0x + x = 0 x(x 0x + = 0 x(x = 0 Thus we get x = 0 and x = as two possible solutions. However, our original equation had domain restrictions. Checking both answers shows that both make the denominator of the rational expression on the left hand side of the equation 0, and thus are not solutions to the equation. Therefore the equation as 0 solutions. 6. The difference quotient of a function f(x is the quotient quotient of f(x = 3 x. f(x + h f(x, h 0. Find the difference h Solution: Recall that f(x + h means that we evaluate the function f at x + h. Thus plugging in x + h where we see x, we get that f(x + h = 3 x+h. Now we can put this together in the difference quotient formula. 3 x+h 3 x 3x x(x+h 3(x+h x(x+h = = h h Combining denominators, and simplifying further, we get 3x (3x+3h x(x+h h = 3x 3x 3h x(x+h h 3x 3x 3h x(x+h h = 3h x(x + hh = 3 x(x + h Therefore the answer is 3 x(x+h. 7. In a warehouse, a stack of 6 foam mattress toppers are piled up. Each mattress topper is originally 3 inches thick. Each is compressed by a third each time an additional mattress topper is piled on top. Rounded to the nearest inch, which of the following is the total height of the pile? Solution: We are finding the sum of heights of mattresses, where each is decreasing in size by /3 each iteration. Thus the very top mattress is the original height of 3, the second is compressed to /3 of 3, or, the third is compressed to /3 of, or 4/3, etc. Using either a partial geometric sum formula, or simply adding together the approximations of the six terms, you get Thus the height is approximately 8 inches. Alternatively, you can make the finite geometric series ( k 3 3 k=0 and use a partial sum formula to arrive at the answer. 8. Find the sum of the solutions to the equation (3x 0(x + = 0 Solution: Even though the left hand side is factored already, it is not set equal to 0. Thus we ll have to multiply out the left hand side, move the 0 over and then solve the resulting quadratic using the zero product property. Page

3 (3x 0(x + = 0 3x 7x 0 = 0 3x 7x 0 = 0 (3x + (x 4 = 0 Therefore there are two solutions x = 3 and x = 4. Thus the solution is = The student council is made up of four sophomores, two juniors and three seniors. A yearbook photographer would like to line up all members of the student council in a line for a picture. How many different pictures are possible if students in the same grade stand beside each other? Solution: Since the students of the same grade need to stand together, we need to arrange four things: Sophomores, Juniors, Seniors and then also the grde levels themselves. That is, there is 4!!3! ways to arrange the students within their groups in the order sophomores, juniors then seniors, but there are 3! ways to arrange each of the groups (seniors first, then juniors, sophomores; seniors first, then sophomores, juniors; etc.. Thus the final answer is 4!!3!3!. 0. For f(x defined piecewise below, evaluate f(f(f(f( 3. x if x < f(x = if x 3 x if x > 3 Solution: To evaluate f(f(f(f( 3, we work our way from the inside outwards. Begin with f( 3. Since 3 <, f( 3 = 3 = 3. We now have f(f(f(3. Since x = 3 falls into the middle condition of the piecewise, then f(3 =. Thus we now have f(f(. Again as <, f( = =. Finally, we have simplified down to f(. As > 3, f( =. Therefore f(f(f(f( 3 =.. Find the horizontal asymptote of the graph of the function f(x = 6x + x + x3 + 7x + 4x + 8x + 6x. Solution: In order to find the horizontal asymptote, you need the combine the terms into a single rational function. Notice that 8x + 6x = (4x + (x. Thus to get a common denominator, we need only multiply the first term by x x. This gives 6x ( + x x + x3 + 7x + 4x + x 8x + 6x = x3 + 6x x 8x + x3 + 7x + + 6x 8x + 6x = 6x + x + 8x + 6x Now that we have a single rational function, the horizontal asymptote is y = a, where a is the ratio of the coefficient of the leading term of the numerator to the coefficient of the leading term of the 6 denominator (since the degrees of the leading terms are the same: 8 =. Thus y = is the horizontal asymptote.. If x + x = 3, then the value of x3 + x 3 is Solution: Notice that since we are wanting the value of x 3 + x 3, we should cube x + x. Expanding ( x + x 3 we get the expression: x 3 +3x+ 3 x + x 3. Cubing the right hand side as well gives the equation Now, rearranging, we get x 3 + 3x + 3 x + x 3 = 7 x 3 + x 3 + 3x + 3 x = 7 As 3x + 3 x = 3 ( x + x, and by assumption, x + x = 3, we have 3x + 3 x this in, we get x 3 + x = 7 == 3 3 = 9. Substituting Page 3

4 and so x 3 + x 3 = Find the slope of the line perpendicular to the line connecting the two points ( 4, 3 and ( 6,. Solution: The formula for the slope of a line between two points is m = y y x x. Thus, the slope is: m = = 4 3 = 4 3 = 6 Now, the question asks for the slope of a line perpendicular to the line connecting the points. The slope of a perpendicular line is the negative reciprocal of the slope of the original line, and so the slope of the perpendicular line is In the senior class at a particular high school, 4 students are taking calculus, students are taking physics, and students are taking both calculus and physics. If there are 00 people in the senior class, what is the probability that a randomly selected student is taking calculus or physics? Solution: To solve this problem, we can either fill out a Venn Diagram, or use the Union Rule. The union rule states that P (E F = P (E + P (F P (E F, for events E and F. Since we are looking for the probability of taking calculus or physics, we are solving for P (E F in the equation. Now, P (taking calculus = 4/00, P (taking physics = /00 and P (taking calculus and physics = /00. Thus P (taking calculus or physics = P (taking calculus + P (taking physics + P (taking calc and physics = 4/00 + /00 /00 = 76/00 Thus P (taking calculus or physics = 76/00 or To which of the following expressions is 8 x x 7 equal to for all real negative values of x? Solution: Recall that n x n = x for even n and n x n = x for odd n. Therefore for all values of 8 x, x8 = x and 7 x 7 = x. But since x < 0 in this question, we know that x = x. Thus 8 x8 + 7 x 7 = x + x = x + x = 0. Therefore the correct answer, for negative values of x, is The graph of the function f is shown below. How many solutions does the equation f(f(x = have? Solution: This problem will involve solving two equations: for which z are f(z = and then for which values of x do f(x = z. To begin with, we can see that z = and z = both have f(z =. For which x do f(x = or f(x =? For f(x =, we have two values of x. It s hard to tell exactly what they are, but approximately, x = 6 and x = 3. both have f(x = and so f(f(x = f( =. For f(x =, we can see there are four total values. Their approximate values are x = 4., x =., x =. and x =., all of which have f(x = and so f(f(x = f( =. Thus there are a total of 6 values for which f(f(x =. 7. Exactly how many solutions are there to the equation x + x 3 =? Solution: Recall that x = x when x 0 and x = x when x < 0. Applying similar logic, we can see that x = x when x and x = (x = x when x < and x 3 = x 3 when x 3 and x 3 = (x 3 = 3 x when x < 3. This breaks the real numbers into three intervals, Page 4

5 (,, [, 3 and [3,. We will solve the equation in each of these three intervals, and that will give us all solutions. For (,, we know that both x = x and x 3 = 3 x. Thus the equation simplifies to x + 3 x =. Further work with this gives x =, which gives a single answer of x =. [Technically this is in the next interval, but we can count it either place. The choice of where to include was arbitrary] Now for the interval [, 3, we know that x = x and x 3 = 3 x, since all values on [, 3 are less than x = 3. Thus this gives the equation x + 3 x =. However, this simplifies to the statement =, which is always true. Thus all values in the interval [, 3 satisfy the equation. Therefore there are infinitely many solutions to this equation and so the answer is None of These. Note: We could finish the problem of finding all solutions by solving the equation on [3,, but as the problem was only asking for the number, we do not need to. 8. If 4 and are the first and fifth terms of an arithmetic sequence, what is the sum of the second, third and fourth terms of the sequence? Solution: Since we are told that 4 is the fifth term of the sequence, we can set up an equation to solve. The five terms of the sequence will be, + x, + x, + 3x, 4 + 4x =. Solving for x in this equation, we get x =. Thus the second, third and fourth terms are 8, 0 and. Their sum is then 30 or. 9. Suppose that f(x = x 6x, g(x = x + 3 and h(x = x. Find K(x, where K(x = (f (g h (x. Solution: To find K(x we will follow the order of composition indicated in the problem. First, we compose (g h(x. Then, Simplifying, we get K(x = x 9. (g h(x = g(h(x = g( x = x + 3 f( x + 3 = ( x + 3 6( x + 3 = x + 6 x x 8 0. Let f be a function satisfying f(xy = f(x for all positive real numbers x and y. If f(00 = 3, what y is the value of f(600? Solution: To find f(600, knowing only the value of f(00 we must write 600 in terms of 00. We can do this as: 600 = Then Thus the value of f(600 is. f(600 = f(00 6 = f(00 = f(00 6/ 6 = 3 6 =. The graph of the function y = x x + 6 x has two vertical asymptotes x = a and x = b. Find ab. + 3x 4 Solution: Vertical asymptotes occur when the denominator is zero, but the numerator is non-zero. Using the quadratic formula, we find the two roots of the polynomial in the denominator are x = 3± 4 4. The question asks for the product, and so = =.. A line with slope 3 intersects a line with slope at the point (4,. What is the distance between the x-intercepts of these lines? Solution: To find the distance between x-intercepts, we first need to find the equations of the two lines. The first line has slope m = 3 going through point (4,. Thus y = mx + b = 3(4 + b = 4 + b b = 7 Page

6 So the equation of the first line is y = 3x 7. Similarly, we can find that the equation of the second line is y = x +. Now, to find the x-intercepts, we set y = 0 and solve. 0 = 3x 7 7 = 3x 9 = x Thus the x-intercept of the line y = 3x 7 is (9, 0. Similarly, we can find the x-intercept of the line y = x + is (44, 0. Finally, the distance between these points is So the answer is 3. d = ( (0 0 = 3 = 3 3. Six houses in a row are each to be painted with one of the colors Red, Blue, Green and Yellow. In how many different ways can the houses be painted so that no two adjacent houses are the same color? Solution: To solve this problem, we will consider for each house how many color options we can use. Then, we ll use the multiplication principle. For the first house, we can use any of the four colors. For the second house, the only house that has been painted is the first, so we only have to avoid that color. Thus the second house has 3 color options. For the third house, only one of its adjacent neighbors has been painted, and so again we only have one color we must avoid. Thus the third house also has 3 color options. Similarly, all of the remaining houses have 3 color options, as they only need to avoid the color of their painted neighbor to the left. Thus we have House House House 3 House 4 House House 6 giving us 4 3 = 97 ways to paint the houses. 4. A fixed point of a function f is a value c in the domain of f such that f(c = c. Find the number of fixed points of the function f(x = x +. Solution: To find the number of fixed points of f(x, we just need to find all c such that f(c = c. This comes down to solving the equation c+ = c. Multiplying both sides by c+ we get = c(c+ = c +c. Rearranging that, we get the c + c = 0. This expression does not easily factor, but either the quadratic formula or completeting the square shows that this equation has two solutions: c = + and c =. Thus there are fixed points for f(x.. Which of the following best describes the graph of the equation 9x 4x = 4y + 6y 9? Solution: The easiest way to tell the shape of such an equation is to get it into standard form. Moving the variables to the left hand side and rearranging gives us 9x 4x 4y 6y = 9. We can complete the square to get it into standard form: 9(x 6x 4(y + 4y = 9 9(x 6x + 9 4(y + 4y + 4 = (x 3 4(y + = 36 (x 3 4 (y+ 9 = This is the standard form of a Hyperbola. 6. The equation x /6 + x / = x /3 has two nonzero real solutions, a and b. Assuming a < b, find a + b. Solution: To begin with, it will help to rewrite the problem with everything moved to one side, and to rewrite the exponents with a common denominator. Doing this, we get the equation x /6 x 4/6 + x 3/6 = 0 Page 6

7 Now, you can see that each of the three terms share an x 3/6. We will factor it out: x 3/6 (x /6 x /6 + = 0 The factor x 3/6 will result in a solution of 0, but since the problem asked for nonzero solutions, we can ignore this factor. This leaves us with a quadratic-like equation of Let z = x /6. Then we can write our equation as x /6 x /6 + = 0 z z + = 0 This factors as (z (z = 0, which gives solutions of z = and z =. Finally, after we substitute x back in, we have to solve the equations x /6 = and x/6 =. Raising both sides to the sixth power gives that x = = 6 64 and x = 6 = 64. Lastly, the problem wants the answer in the form a + b, where a is the smaller of the two solutions, which gives us = Suppose f( = 3 and for all integers n >, f(n = f(n + if n is even and f(n = f(n if n is odd. Find f(. Solution: To find f(, we iterate until we can use f( = 3 to get a final answer. f( = f( Since is odd, we use the rule for odd numbers = f(4 = (f(4 + Since 4 is even, we use the rule for even numbers = (f(3 + = (f(3 + Replace f(3 with f(3 = (f( + = ([f( + ] + = ([f( + ] + = ([3 + ] + Replace f( with 3, the given = ([4] + = (9 = 8 Thus f( = Find the remainder when the polynomial f(x = x 4 + x 3 3x 8 is divided by the polynomial x +. Solution: There are several ways to do this problem: Long division, synthetic division, or the Remainder Theorem. The Remainder Theorem states that the remainder when you divide a polynomial P (x by a linear factor x a is just P (a, the polynomial evaluated at a. In our case, the a = [as x + = x ( ], and so the remainder we are finding is simply f( = ( 4 + ( 3 3( 8 = 0. Therefore the answer was None of These. 9. On an island, 99% of the population are natives. Some natives emigrate so that only 98% of the population are natives. If the initial population of the island was 000, how many natives emigrated? Solution: If the initial population is 000, and 99% of those are natives, then the breakdown at the beginning is 99% 000 = 990 natives and = 0 non-natives. Some natives then emigrated. However, no non-natives emigrated. Thus the 0 non-natives, all of whom are still present, are now % of the population (since 98% is now native. We can then set up the equation: % p = 0 where p is the new total population. Solving this equation gives p = 00. Therefore the new population of the island is 00, and thus the population of the island decreased by 00 people. However, all who emigrated were native, and thus 00 natives emigrated. Page 7

8 30. Find the inverse function f (x of the function f(x = x + x + 3. Solution: One method of finding an inverse function is to swap x and y in the original function, and then re-solve for y. Thus we have x = y+ y+3 Swap x and y x(y + 3 = y + Multiply both sides by y + 3 xy + 3x = y + Distribute xy y = 3x + Rearrange y(x = 3x Factor out a y y = 3x x Divide both sides by x Thus the inverse is f (x = 3x x. Caution: f (x is the function which composes with f(x to yield the identity. That is, f(f (x = f (f(x = x. In general, it is not the same as f(x, the reciprocal of the function. 3. Find the product of all real numbers k for which the function f(x = kx + x + k touches, but does not cross, the x-axis. Solution: The are multiple ways to approach this problem. One is to realize that f(x is a parabola. If a parabola is to touch, but not cross the x-axis at a point, then it must have a double root at that point. That is, it must be able to be written as a square. If you are to square a linear term ax + b and get kx and k as two of the terms, we must have it factor as kx + x + k = ( kx ± k. Expanding the square gives two equations: x = k and x = k. Solving these equations gives k = and k = and so the product is - 4 Another approach: First, we factor f(x into standard form. As it touches but does not cross the x-axis, it will factor as f(x = a(x h. Expanding this we get f(x = ax ahx + ah. However, we also know that f(x = kx + x + k. Equating coefficients of terms of the same degree, we get three equations: a = k, ah = and ah = k. Combining the first and third equations, we can conlcue that h = and so h = ±. We can then substitute these into the equation ah = and conclude that a = ±. Thus the product is 4 3. The graphs of equations 9x + y = 9 and 3x + y = 6 intersect at points (a, b and (c, d. Find a + b + c + d. Solution: We need to solve the non-linear system: { 9x + y = 9 3x + y = 6 To do this, we can solve for y in the linear equation, and then substitute it into the other equation. (You could also solve for x, but then you would have to substitute that into 9x, which requires a little more work that substituting into y. Solving for y gives y = 3 3 x. 9x + y = 9 9x + ( 3 3 x = 9 9x + 9 9x x = 9 x + x + 4 x = 4 x x + = 4 x x = 0 x ( 4 x = 0 Thus x = 0 and x = 4. To find the y values, we can substitute back into either equation. We ll use the one that is already solved for y: y = 3 3 (0 ( = 3 y = = 9 Page 8

9 Thus the answer is = 8 ( 33. Let f(x be a function such that f(x + f x = x for all x not equal to 0 or. What is the value of f(? ( Solution: To find f(, plug x = into the given equation to get f( + f to f( + f( = =, which simplifies However, we do not ( know the value of f(. To find it, we can plug x = into the given equation to get f( + f =, which simplifies to ( f( + f ( = This again presents a problem that we do not know the value of f (, and so we can try again to plug x = into the given equation. This gives us f ( ( + f / =. Simplifying, we get f ( + f( = This gives us a system of three equations in three unknowns: f( + f( = f( + f ( = f ( + f( = If you multiply the second equation through by and then add all three equations, you get f( = 7 and so f( = The following transformations are applied (in the given order to the graph of y = x. I. Reflection about the x-axis II. Horizontal shift left 3 units III. Vertical shift up unit Determine the equation of the graph produced as a result of applying these transformations. Solution: To first reflect about the x-axis, multiply by, giving the equation y = x. To shift left by 3 units, we replace x by x + 3, which gives y = x + 3. Finally to shift up by unit, we add, which gives y = x Thus the answer was None of These. 3. Find the sum of all solutions (both real and complex to the equation ix + 7x i = 0, where i =. Solution: Despite the complex coefficients, you can solve this equation just like any other quadratic. Let s solve by factoring. If you like, you can first factor out an i to get i(x 7ix = 0 However, we can also factor as (ix+4(x 3i = 0. Solving these two linear equations gives x = 4 i and x = 3i. Thus the sum of all solutions is 7i. = 4i Page 9

11 4. Which of the following functions has a graph with exactly one hole, occurring when x = and exactly one vertical asymptote, occurring when x = 7? Solution: Recall that a hole occurs at x = a if a is a zero of the denominator and the numerator. An asymptote occurs at x = b if b is a zero of the denominator but not the numerator. Systematically factoring each of the answer choices shows that: (a has a hole at x = and an asymptote at x = 7, but also has a hole at x = 0; (b has a hole at x = 7 and an asymptote at x = ; (c is the correct answer, with a hole at x = and an asymptote at x = 7; (d has holes at x = 7 and x = 0 with an asymptote at x =. Thus the correct answer is (c f(x = x4 x 3 x +6x Let S = n, for n a positive integer. How many factors of are there in the prime factorization of S? Solution: We can use the summation formula to make this sum more compact: S = 0n (0 n +. The number 0 n is even, as it is divisible by, and 0 n + is odd. To see the number of factors of in 0 n, we first will factor it: 0 n = ( n = n n Thus 0 n has n factors of. After dividing by, we have that S has n factors of. 44. Amelia is driving at 60 mph, tapping her toe to a song at a rate of taps per 3 seconds. She notices that every 0th tap always falls on a telephone pole. Assuming the telephone poles are always an equal distance apart on this road, how many miles apart are they? Solution: Every 0th tap is every 6 seconds, since we have taps every 3 seconds. Going 60mph means 60 miles takes hour, which equates to one mile in one minute. Since 6 seconds is 0th of a minute, the poles are every 0 miles. 4. Find the value of k so that the function f(x is continuous. { kx f(x = + 3 x < k x Solution: Informally, for a function to be continuous, it needs to be able to be graphed without any holes, jumps, vertical asymptotes or other discontinuities of the graph. In order to avoid a jump at x =, we need both pieces evaluated at x = to be the same value. That is, we need kx +3 = k, when x =. This gives the equation 4k + 3 = k. Solving for k, we get k =, which is None of These. More precisely, we require the limit from the left of x = to be equal to the limit from the right of x = and both to be equal to f( = k. That is, lim x f(x = lim +f(x = f( = k 46. Let (a, b be a point on the graph of 6x + 36y 8x + 48y = 7. What is the largest such b? x Solution: In an ellipse, the largest y value will be directly above the center. So first, we find the center by completing the square. 6x 8x + +36y + 48y = 7 6(x x + 36(y + 4 3y = 7 6(x x (y = (x (y + 3 = 44 (x (y+ 3 4 = Thus the center is ( 4, 3. The top-most y value on the graph will occur directly above the center, along the minor axis, which has length b = 4 = 4. So the distance from the center along the minor axis to the curve is. Thus b = 3 + = 4 3. Page

12 47. Divide 7i by 6 i. Put your answer in a + bi form. Solution: To get the answer into a + bi form, we must multiple both the numerator and denominator by the complex conjugate, 6 + i, of the denominator. 7i 6 i Thus the answer is i. 48. Find the sum of all solutions to the equation ( 6 + i 6 40i 4i 6 40i i = 6 + i 36 4i = = = i (x 3 + x + x (4x 4x + 6 = 0. Solution: Notice that this equation is the difference of two squares. Recall that for any a, b, a b = (a + b(a b. We can factor and simplify the equation: ( ( (x 3 + x + x + (4x 4x + 6 (x 3 + x + x (4x 4x + 6 ( ( = 0 ( x 3 + x + x + ( x 3 3x + 9x 7 = 0 x (x + + (x + x (x 3 + 9(x 3 = 0 (x + (x + (x + 9(x 3 = 0 The solutions to the four factors are, respectively, i, i,, 3i, 3i, 3. Finally, the sum of all solutions is. 49. Two committees consisting of 3 and people, respectively, are to be formed from a group of 8 people. Assuming no person is on both committees, in how many ways can this be done? Solution: Since we are not putting anyone on multiple committees, after we choose 3 people from the 8, all of the remaining will be forced to be on the second committee, so we need only account for the choice of the 3 person committee. The formula for this is Therefore the answer is 8! 3!!. 0. Which of the following expressions is equivalent to ( 8 = 8! 3 3!! log (x 3 log (xy + 3 log (y /? Solution: First, using the Power Rule, move the constant multiple on the third term to the power to get 3 log (y / = log (y 3/. Then, we can combine from left to right using the Quotient and Product rules: ( x log (x 3 log (xy + log (y 3/ 3 y 3/ ( x = log xy = log Thus the final answer is log ( x y /. y / Page

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