Precalculus Fall 204 Midterm Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find an equation in standard form for the hyperbola that satisfies the given conditions. ) Foci at (±8, 0), transverse axis with length 4 ) A) x 2 49 - y 2 64 = B) x 2 49 - y 2 = C) x 2 - y 2 49 = D) x 2 64 - y 2 49 = 2) Center (-2, 4), focus (-2, 0), vertex (-2, 6) 2) A) (y + 4) 2 - (x + 2) 2 = B) (y + 4) 2 + (x + 2) 2 = 4 2 4 2 C) (y - 4) 2 4 - (x + 2) 2 40 = D) (y - 4) 2 4 - (x + 2) 2 2 = ) Foci at (0, ±), a conjugate axis with length 0 ) A) y 2 2 - x 2 49 = B) y 2 49 - x 2 00 = C) y2 00 - x 2 49 = D) y 2 24 - x 2 2 = 4) Vertices at (±2, 0), foci at (±, 0) 4) A) x 2 49 - y 2 4 = B) x 2 4 - y 2 49 = C) x 2 4 - y 2 4 = D) x 2 4 - y 2 4 = Determine the intervals on which the function is increasing, decreasing, and constant. ) ) A) Increasing on (-, ); Decreasing on (-, -) B) Increasing on (-, ); Decreasing on (, ) C) Increasing on (-, -); Decreasing on (-, ) D) Increasing on (, ); Decreasing on (-, )
6) 6) A) Increasing on (, ); Decreasing on (-2, 0) and (, ); Constant on (2, ) B) Increasing on (-2, 0) and (, 4); Decreasing on (-, -2) and (, ) C) Increasing on (-, 0) and (, ); Decreasing on (0, ); Constant on (-, -) D) Increasing on (-2, 0) and (, ); Decreasing on (, ); Constant on (-, -2) Find the vertex, focus, directrix, and focal width of the parabola. ) x = 8y2 ) A) Vertex: (0, 0); Focus: 0, 2 ; Directrix: y = - ; Focal width: 2 2 B) Vertex: (0, 0); Focus: C) Vertex: (0, 0); Focus: D) Vertex: (0, 0); Focus: 8, 0 ; Directrix: x = - ; Focal width: 0. 8 2, 0 ; Directrix: x = ; Focal width: 2 2 2, 0 ; Directrix: x = - ; Focal width: 0. 2 8) x2 = 28y 8) A) Vertex: (0, 0); Focus: (, 0); Directrix: x = ; Focal width: B) Vertex: (0, 0); Focus: (, 0); Directrix: y = ; Focal width: 2 C) Vertex: (0, 0); Focus: (0, ); Directrix: y = -; Focal width: 28 D) Vertex: (0, 0); Focus: (0, -); Directrix: x = -; Focal width: 2 9) (x - 8)2 = 6(y - 6) 9) A) Vertex: (6, 8); Focus: (22, 8); Directrix: x = -8; Focal width: 6 B) Vertex: (6, 8); Focus: (0, 8); Directrix: x = 4; Focal width: 4 C) Vertex: (-8, -6); Focus: (-8, 0); Directrix: y = -22; Focal width: 6 D) Vertex: (8, 6); Focus: (8, 0); Directrix: y = 2; Focal width: 6 0) y2 = -24x 0) A) Vertex: (0, 0); Focus: (0, -6); Directrix: y = 6; Focal width: 96 B) Vertex: (0, 0); Focus: (-6, 0); Directrix: x = 6; Focal width: 24 C) Vertex: (0, 0); Focus: (6, 0); Directrix: x = -6; Focal width: 6 D) Vertex: (0, 0); Focus: (-6, 0); Directrix: y = 6; Focal width: 96 2
Find the domain of the given function. x ) f(x) = x2 + x A) (-, 0) (0, ) B) (-, 0) (0, ) (, ) C) (-, -) (-, 0) (0, ) D) (-, -) (-, ) (x + )(x - ) 2) f(x) = x2 + 9 A) (9, ) B) (-,-9) (-9,9) (9, ) C) All real numbers D) (-,) (-,) (, ) ) 2) ) f(x) = 9 - x ) A) All real numbers B) (-, 9] C) ( 9, ) D) (-,9) (9, ) Solve the equation. 4) x + - x - = 8 x2-2 ) A) x = -4 B) x = 68 C) x = 4 D) x = 2 6x x - 6-4 x = 24 x2-6x 4) ) A) x = 2 or - 2 B) x = or - C) x = 2 D) x = 2 Find the zeros of the function. 6) f(x) = x + 6x2 + 8x + 40 6) A) -, -, and -4 B) 0,,, and 4 C) -, -, and -4 D),, and 4 ) f(x) = x + x2 + 2x ) A) - 2 and B) - 2 and - C) 0, - 2, and D) 0, - 2, and - Find the eccentricity of the ellipse. 8) x2 + y2 = 8) A) 0 B) 2 C) i 2 D) 2
Solve the problem. 9) A satellite is to be put into an elliptical orbit around a moon. The moon is a sphere with radius of 6 km. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 900 km to 69 km. 9) A) C) x2 69 + y 2 900 = B) x2 62 + y2 462 = x2 462 + y 2 62 = D) x2 900 + y 2 69 = 20) Let f(x) compute the time in hours to travel x miles at 9 miles per hour. What does f-(x) compute? A) The miles traveled in 9 hours B) The miles traveled in x hours C) The hours taken to travel x miles D) The hours taken to travel 9 miles 20) 2) The roof of a building is in the shape of the hyperbola y2 - x2 = 46, where x and y are in meters. Refer to the figure and determine the height h of the outside walls. 2) a = b = 6 m A) 40 m B) 9. m C) -0 m D) 82 m 22) An elliptical riding path is to be built on a rectangular piece of property that measures 8 mi by 6 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides 22) A) x 2 64 + y 2 9 = B) x 2 6 + y 2 6 = C) x 2 9 + y 2 64 = D) x 2 9 + y 2 6 = 4
2) If the average cost per unit C(x) (in dollars) to produce x units of plywood is given by C(x) = 200 x + 40, what is the unit cost for 40 units? A) $.00 B) $0. C) -$0.00 D) $0.00 24) A domed ceiling is a parabolic surface. For the best lighting on the floor, a light source is to be placed at the focus of the surface. If m down from the top of the dome the ceiling is 8 m wide, find the best location for the light source. A).2 m down from the top B) 2. m down from the top C) 0.6 m down from the top D).8 m down from the top 2) 2) 24) 2) 6 m 6 m A tunnel is in the shape of a parabola. The maximum height is 6 m and it is 6 m wide at the base. What is the vertical clearance m from the edge of the tunnel? A) 0.6 m B).4 m C) 0. m D). m 26) Suppose a cost-benefit model is given by y =.8x, where y is the cost in thousands of dollars for 00 - x removing x percent of a given pollutant. Find the cost of removing % to the nearest dollar. A) $8 B) $0 C) $800 D) $2046 26) 2) A comet follows the hyperbolic path described by x 2 2 - y 2 8 the sun is the focus of the path, how close to the sun is the vertex of the path? =, where x and y are in millions. If 2) A) 4 million B) 6.6 million C) million D).6 million 28) Sue invested $0,000, part at.9% annual interest and the balance at 6.6% annual interest. How much is invested at each rate if a -year interest payment is $64.0? A) $80 at.9% and $690 at 6.6% B) $0 at.9% and $6290 at 6.6% C) $6290 at.9% and $0 at 6.6% D) $60 at.9% and $480 at 6.6% 28) 29) The profit made when t units are sold, t > 0, is given by P = t2-2t + 22. Determine the number of units to be sold in order for P < 0 (a loss is taken). A) t = 4 or t = 8 B) t > 0 C) 4 < t < 8 D) t < 4 or t > 8 29)
0) Let f(x) compute the cost of a rental car after x days of use at $44 per day. What does f-(x) compute? A) The number of days rented for x dollars B) The cost of rental for 44 days C) The cost of rental for x days D) The number of days rented for 44 dollars 0) ) Find the remainder when x44 - is divided by x +. ) A) 4 B) 6 C) -4 D) -6 Find the center, vertices, and foci of the ellipse with the given equation. 2) (x + ) 2 + (y + 4) 2 = 2) 2 9 A) Center: (-, -4); Vertices: (-8, -4), (2, -4); Foci: (-, -4), (, -4) B) Center: (-, -4); Vertices: (-4, -8), (-4, 2); Foci: (-4, -), (-4, ) C) Center: (-, -4); Vertices: (-8, -4), (2, -4); Foci: (-6, -4), (0, -4) D) Center: (-, -4); Vertices: (-4, -8), (-4, 2); Foci: (-4, -6), (-4, 0) ) x2 + 8y2 = 24 ) A) Center: (0, 0); Vertices: 0, -8, 0, 8 ; Foci: 0, -, 0, B) Center: (0, 0); Vertices: 0, -2 2, 0, -2 2 ; Foci: 0, -, 0, C) Center: (0, 0); Vertices: -2 2, 0, -2 2, 0 ; Foci: -, 0,, 0 D) Center: (0, 0); Vertices: -8, 0, 8, 0 ; Foci: -, 0,, 0 4) x2 00 + y 2 6 = 4) A) Center: (0, 0); Vertices: (0, -0), (0, 0); Foci: (0, -6), (0, 6) B) Center: (0, 0); Vertices: (-0, 0), (0, 0); Foci: (-6, 0), (6, 0) C) Center: (0, 0); Vertices: (0, -0), (0, 0); Foci: (0, -8), (0, 8) D) Center: (0, 0); Vertices: (-0, 0), (0, 0); Foci: (-8, 0), (8, 0) Solve the polynomial inequality. ) x4-6x - 9x2 + 264x + 260 0 ) A) (-, -) [6, 0] B) [-, -] (6, 0) C) [-, -] [6, 0] D) [-, 0] 6) (x + 4)(x + )(x - 4) > 0 6) A) (-4, -) (4, ) B) (-, -) C) (4, ) D) (-, -4) (-, 4) Find the (x,y) pair for the value of the parameter. ) x = 9t + and y = 9-2t for t = ) A) (6, ) B) (9, -2) C) (9, 9) D) (, 6) 8) x = t and y = t2-9 for t = 4 8) A) (2, 6) B) (2, ) C) (, 6) D) (6, 4) Find a cubic function with the given zeros. 9) -, 6, - 9) A) f(x) = x + 6x2 - x + 20 B) f(x) = x + 6x2 - x - 20 C) f(x) = x - 6x2 - x - 20 D) f(x) = x + 6x2 + x - 20 6
40) 2, - 2, 40) A) f(x) = x - x2 + 2x + 4 B) f(x) = x + x2-2x + 4 C) f(x) = x - x2-2x + 4 D) f(x) = x - x2-2x - 4 Find the asymptote(s) of the given function. 4) f(x) = x - vertical asymptotes(s) 4) x2-6 A) x = -4 B) x = 4 C) x = 4, x = -4 D) x = 42) g(x) = x 2 + x - 2 x - 2 horizontal asymptotes(s) 42) A) None B) y = - C) y = 8 D) y = 2 4) f(x) = x - vertical asymptotes(s) 4) x2 + 2 A) None B) x = 2 C) x =, x = - D) x = -2 Give the equation of the function g whose graph is described. 44) The graph of f(x) = x is vertically stretched by a factor of 4.8. This graph is then reflected across the x-axis. Finally, the graph is shifted 0. units downward. A) g(x) = 4.8 -x - 0. B) g(x) = 4.8 x - 0. C) g(x) = 4.8 x - 0. D) g(x) = -4.8 x - 0. 44) 4) The graph of f(x) = x2-4x + is horizontally shrunk by a factor of /4. 4) A) g(x) = 4x2-6 x + B) g(x) = 4x2-6 x + 2 C) g(x) = 4 x 2 - x + 4 D) g(x) = 6 x 2 - x + Find an equation in standard form for the ellipse that satisfies the given conditions. 46) Vertices at (±0, 0) and foci at (±, 0) 46) x A) 0 + y 2 = 2 B) x2 62 + y 2 0 = C) x2 00 + y 2 = D) x2 00 + y = 4) Minor axis endpoints (±2, 0), major axis length 24 4) A) x 2 2 + y 2 2 = B) x 2 44 + y 2 4 = C) x 2 2 + y 2 2 = D) y2 44 + x 2 4 = 48) The horizontal major axis is of length 8, and the minor axis is of length 8. 48) A) x 2 9 + y 2 4 = B) x 2 6 + y 2 8 = C) x 2 4 + y 2 9 = D) x 2 8 + y 2 6 =
Find the inverse of the function. 49) f(x) = x - 2 49) A) f-(x) = x + 2 B) f-(x) = x - 2 C) f-(x) = x + 2 D) Not a one-to-one function 0) f(x) = -4x - 4 x + 8 A) Not a one-to-one function B) f(x) = -8x - 4 x + 4 C) f-(x) = x + 4-8x - 4 D) f-(x) = -4x - 4 x + 8 0) ) f(x) = x - ) A) f-(x) = x + B) f-(x) = x - C) Not a one-to-one function D) f-(x) = x + For the given function, find all asymptotes of the type indicated (if there are any) x + 2 2) f(x) =, slant 2) x2 + 9x - 8 A) y = x + 2 B) y = x + C) None D) x = y + ) f(x) = x -, vertical ) x2 - A) x = B) x = C) x = - D) x =, x = - Match the function with the graph. 4) 4) A) g(x) = -(x - 2)2 B) g(x) = -x2-2 C) g(x) = (x + 2)2 D) g(x) = -x2 + 2 8
) ) A) y = x - B) y = x + C) y = x - + D) y = x - State the domain of the rational function. 6) f(x) = x - x2 - A) (-, ) (, ) B) (-, - ) (-, ) (, ) C) (-, ) (, ) D) (-, -) (-, ) 6) ) f(x) = 4 - x A) (-, ) (, ) B) (-, 4) (4, ) C) (-, - ) (-, ) (, ) D) (-, - 4) (- 4, 4) (4, ) ) Determine whether the formula determines y as a function of x. 8) y = 9x + 8) A) No B) Yes 9) x = y2 + 9) A) No B) Yes Find a direct relationship between x and y. 60) x = 4t and y = 9t + 60) A) y = 9 4 x + B) y = 6x + C) y = 6x D) y = x 4 6) x = t - and y = t2-6t 6) A) y = x2-6x + B) y = x2 + 4x - C) y = x2 + x + 0 D) y = x2 + x + 20 Solve the inequality. 62) 2x - x + < 0 62) A), /2 B) -, /2 C) /2, D) /2, 9
6) x 2 x - 2 < 0 6) x + 4 A) -4, 0 0, B) -4, 0 2, C) -4, D) -4, 0 0, 2 Use an equation to solve the problem. 64) If Gloria received a 0% raise and is now making $26,400 a year, what was her salary before the raise? A) $24,400 B) $2,000 C) $2,400 D) $24,000 6) One positive number is twice another positive number. The sum of the two numbers is 60. Find the two numbers. A) 20, 22 B) 60, 00 C) 80, 80 D) 20, 240 64) 6) Find the vertices and foci of the hyperbola. 66) (y + 4) 2 - (x + 2) 2 = 66) 64 6 A) Vertices: (2, -2), (-0, -2); Foci: (-0, -2), (2, -2) B) Vertices: (-2, 4), (-2, -2); Foci: (-2, 6), (-2, -4) C) Vertices: (4, -2), (-2, -2); Foci: (6, -2), (-4, -2) D) Vertices: (-2, 2), (-2, -0); Foci: (-2, -0), (-2, 2) 6) x 2 4 - y 2 60 = 6) A) Vertices: (0, ± 2); Foci: (0, ±8) B) Vertices: (± 8, 0); Foci: (± 2, 0) C) Vertices: (0, ± 8); Foci: (0, ± 2) D) Vertices: (± 2, 0); Foci: (± 8, 0) Solve the equation algebraically. 68) v2 + 2 = 8-4v2 68) A) ± 4 B) ± 6 C) ± 2 D) ± 6 69) x2 - x - = 0 69) A) - 2 ± 2 4 B) 2 ± 2 9 C) ± 4 D) 2 ± 2 4 Find all rational zeros. 0) f(x) = 4x - 6x2 - x + 4 0) A), -, 4 B) 2, - 2, 4 C) 2, -2, 4 D) 2, - 2, -4 ) f(x) = x + 4x2-2x - 90 ) A) -4, -6, 0 B) 4, 6, -0 C), 6, - D) -, -6, Find the range of the function. 2) f(x) = (x - 2)2 + 2 2) A) (-, ) B) [0, ) C) [2, ) D) (-,2) 0
Find the standard form of the equation of the parabola. ) Vertex at the origin, opens to the right, focal width = 4 ) A) y2 = 4x B) x2 = 4y C) y2 = -4x D) y2 =.x 4) Vertex at the origin, focus at (0, 9) 4) A) y = 6 x 2 B) y = 9 x 2 C) y2 = 6x D) y2 = 9x ) Focus at (-, 2), directrix x = - ) A) (y - 2)2 = 6(x + ) B) (y - 2)2 = 6(x + ) C) (x - 2)2 = 6(y + ) D) (x + )2 = 6(y - 2) Find the vertex, the focus, and the directrix of the parabola. 6) y2-8x - 6y + = 0 6) 9 A) Vertex: - 4, ; Focus: 8, ; Directrix: y = 8 B) Vertex:, ; Focus: (9, ); Directrix: y = C) Vertex:, ; Focus: (, ); Directrix: x = - D) Vertex: 0, ; Focus: (-, ); Directrix: y = ) x2 + 2x - 8y - 9 = 0 ) A) Vertex: -, ; Focus: (-, ); Directrix: y = B) Vertex: -, - 9 8 ; Focus: (-, ); Directrix: y = - 4 8 C) Vertex: -, - ; Focus: (-, -); Directrix: y = - D) Vertex: -, - 4 ; Focus: (-, -); Directrix: y = - Perform the requested operation or operations. 8) f(x) = x + ; g(x) = x - Find f(g(x)). A) f(g(x)) = 9x + 0 B) f(g(x)) = 9x + 6 C) f(g(x)) = 9x + 2 D) f(g(x)) = 9x + 8 8) 9) f(x) = 6x + 8; g(x) = 4x -, find f(g(x)). 9) A) f(g(x)) = 24x + 2 B) f(g(x)) = 24x + 4 C) f(g(x)) = 24x + D) f(g(x)) = 24x + Find the eccentricity of the hyperbola. 80) 8x2-9y2 = 2 80) A) 2 B) C) D) 2 8) x2 - y2 = 49 8) A) 2 B) C) D) 0