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Final exam review #1-2: find an equation for the line with the given properties. Express your answer in slope intercept form. 2 1) Slope = 3; containing the point (6,5) 2) Passing through the points (3,1) and (4,5) #3-4: Find the x and y-intercepts. 3) 𝑦 = 𝑥 2 −5𝑥−6 𝑥+1 4) x2 + y2 = 121 #5-8: find the domain and range. 5) 6) 7) 8) Final exam review #9-11: Use algebra to find the domain of each function. Write your answer in interval notation 𝑥−4 9) 𝑓(𝑥) = 𝑥 2 +6𝑥−7 10) 𝑓(𝑥) = √𝑥 + 5 11) f(x) = 2x – 6 #12: Find the following: a) the interval(s) where the function graphed is increasing b) the interval(s) where the function graphed is decreasing c) The values of x (if any) where the function has a local maximum d) The local maximum value (if any) e) The values of x (if any) where the function has a local minimum f) The local minimum values (if any) 13) Find the average rate of change of f(x) = x3 + 6x2 from 0 to 2 #14-16: describe how the graph of the given function relates to the graph of a common function 14) f(x) = (x+2)2 - 4 15) f(x) = −√𝑥 − 3 – 5 16) 𝑓(𝑥) = |𝑥 − 3| + 4 Final Exam Review 17) f(x) = (x – 3)2 + 4 a) describe the transformation as compared to the function f(x) = x2, specifically state if the graph is shifted left, right, up, down and if any reflection has occurred b) make a table of values and sketch a graph c) state the domain and range of the function d) state the intervals where the function in increasing and decreasing e) state if the function has a local maximum, if it does state the local maximum value f) state if the function has a local minimum, if it does state the local minimum value 18) Use synthetic division to factor the polynomial, then solve the equation. x3 + x2 + 2x + 2= 0 #19 – 20: For each problem find the following: a) Domain b) Vertical Asymptote (if any) c) Horizontal asymptote, or slant asymptote d) x- intercept(s) if any e) y-intercept(s) if any f) Sketch a graph of the function : label all the features found in parts b - e 19. f ( x ) 2x 6 x3 20. f ( x) x 1 x 4 x 21 2 #21-22: Write each side with the same base then solve. Be sure to check your answer. 21) 3x-3 = 27 1 𝑥−2 22) (2) 1 =8 #23-26: Solve the logarithmic equations, round to 2 decimals when needed. 23) log2(x+1) = 5 24) log4(x-5)=3 25) log2 (x+14) – log2 (x+6)= 1 26) log2 (x+2)+log2 (x+6) = 5 Final Exam Review 27) Solve using the substitution method 28) Solve using the elimination method. 4 1 x y 5 5 4 x y 1 2 x 3 y 13 5 x 4 y 21 29) Solve each system of equations, by hand without matrices 2𝑥 + 4𝑦 − 5𝑧 = −5 −𝑥 + 𝑦 + 2𝑧 = 7 𝑥 − 3𝑦 + 3𝑧 = 4 (pair the middle equation with the other 2 and drop out the x’s) #30-31: Solve the following systems of equations. 30) x 7 y 4 x y 10 2 2 31) x 2 y 11 x 2 y 13 32) Find the difference quotient for the function f(x) = x2 +3x - 5: 𝑓(𝑥 + ℎ) − 𝑓(𝑥) ℎ Answers: 2 1) 𝑦 = 3 𝑥+1 2) y = 4x – 11 3) x-intercepts (-1,0) and (6,0) y-intercept (0,-6) 4) x-intercepts (11,0) and (-11,0) y-intercepts (0,11) and (0, -11) 5) domain {3,4,5,7} Range {-1,2,6} 7) 𝑑𝑜𝑚𝑎𝑖𝑛 (−∞, 0] 𝑟𝑎𝑛𝑔𝑒 (−∞, 2] 6) Domain [0,5] range [-5,4] 8) 𝑑𝑜𝑚𝑎𝑖𝑛 (−∞, ∞) 𝑟𝑎𝑛𝑔𝑒 (−∞, ∞) 9) domain (−∞, −7) ∪ (−7,1) ∪ (1, ∞) 10) 𝑑𝑜𝑚𝑎𝑖𝑛 [−5, ∞) 11) 𝑑𝑜𝑚𝑎𝑖𝑛 (−∞, ∞) 12a) (−∞, −3) ∪ (−1, ∞) 12b) (-3,-1) 12e) x = -1 13) average rate of change = 16 12f) y = -12 12c) x = -3 12d) y = -8 14) same shape as g(x) = x2, except moved down 4 units and left 2 units 15) same shape as g(x) = √𝑥, except moved down 5 units, right 3 units and reflected over x-axis 16) same shape as g(x) = |𝑥|, except shifted up 4 units and right 3 units 17a) The graph has the same shape as g(x) = x2, except it is shifted right 3 units and up 4 units. 17b) x 5 4 3 2 1 f(x) or y 8 5 4 5 8 Computation, use calculator to get y - column (5-3)2 + 4 (4-3)2 + 4 (3-3)2 + 4 (2-3)2 + 4 (1-3)2 + 4 17c) Domain (−∞, ∞) Range [4, ∞) (see me for help if you need some finding the domain and range) 17d) The graph is increasing (3, ∞) and decreasing from (−∞, 3) 17e) The graph does not have a high point so it has no local maximum 17f) The low point is the local minimum. We say there is a local minimum at x = 3 and the local minimum value is y = 4 x = -1, ±𝒊√𝟐 18) 19a) all real numbers except 3 19d) (-3,0) 19b) x = 3 19c) y = 2 19e) (0, -2) 19f) y 8 6 4 2 x -18 -16 -14 -12 -10 -8 -6 -4 -2 2 (-3,0) 4 6 8 10 12 14 16 18 -2 (0,-2) -4 -6 -8 20a) all real numbers except -7,3 20b) x = -7 and x = 3 20c) y = 0 (the x-axis) 11d) (1,0) 20e) (0, 1/21) 20f) y 4 3 2 1 x -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 2 3 4 5 6 7 8 9 10 11 12 13 14 21) x = 6 22) x = 5 23) x = 31 24) x = 69 25) x = 2 26) x = 2 27) (5,4) 28) (5,1) 29) (1,2,3) 30) (-79/25, 3/25) and (3,1) 32) 2x + h+ 3 31) (-5/2, 27/4) and (3,4)