e. some other answer 6. The graph of the parabola given below has an axis of symmetry of: a. y = 5 b. x = 3 c. y = 3 d. x = 5 e. Some other answer.

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1 Intermediate Algebra Solutions Review Problems Final Exam MTH 099 December, True or False: (a + b) = a + b. True or False: x + y = x + y. True or False: The parabola given by the equation y = x 5x 6 will have two real x-intercepts.. True or False: The base of an exponential can be negative. 5. The slope of the line between the points: (5, ) and (, ) is: m = y y1 x x 1 = ( ) 5 = = a. 1 1 b. c. d. e. some other answer 6. The graph of the parabola given below has an axis of symmetry of: a. y = 5 b. x = c. y = d. x = 5 e. Some other answer. x 7. The rational expression, x 5x+6 is undefined for what values: can not have x 5x + 6 = 0 (x )(x ) = 0 x, x a. x =, x = b. x = 6, x = 1 c. x =, x = d. x = 6, x = 1 8. What value for x makes the ordered pair (x, 1) a solution to y = x + ( 1) = x + = x + 6 = x x = a. x = b. x = 0 c. x = d. x = 1 e. some other answer 9. True or False: The ellipse given by the equation x the y-axis. + y Write the equation of a circle centered at the origin with a radius of 5. x + y = 5 x + y = 5 = 1 has the major axis along 11. Write the line y x = 1 in slope-intercept form, identify the slope and the y-intercept. y = x + 1 y = x + 1 { slope: y = 1 1 x + y intercept : 1

2 1. Find the equation of the line passing through the points (, ) and (, ). Use y = mx + b where m = ( ) = 1 = y = x+ b use either point to find b, let s use, (, ) which means x = and y = = ( ) + b = + b = b The equation of the line is: y = x 1. Solve the following system of equations: x + y = 1 x = 5 y From x + y = 1 x = 1 y substitute this into the bottom equation for x. (1 y) = 5 y y = 5 y 8 = y = y x = 1 = Solution is : (, ) 1. Solve the following system of equations: x + y = x y = 5 multiply all terms of bottom equation by to make the y-coefficients opposite. x + y = 6x y = 10 x = 1 x = need y () + y = y = Solution is :(, ) 15. A tour group is split into two groups when waiting in line for food at the zoo snack shack. The first group bought 7 slices of pizza and 5 sodas for $8.50. The second group bought 6 slices of pizza and sodas for $.98. How much does one slice of pizza cost and one soda cost? Let P represent the cost of 1 slice of pizza. Let S represent the cost of 1 soda. Total cost for group one: 7P + 5S = 8.50 Total cost for group two: 6P + S =.98 Multiply to top equation by (-) and bottom equation by 5, this forms opposite coefficients on the S variable,then add these equations The cost of one slice is : $.95 The cost of one soda is: $ P 0S = 11 0P + 0S = P = 5.9 P = Simplify the following rational expressions by performing the indicated operation. (a) x+16 x +x 8 = (x+) (x+)(x ) = x (b) x+ x 8 x 8x x+6 = x+ (x ) x(x ) (x+) = x

3 (c) x+ x LCD: x(x + ) x x(x+) (x+) x(x+) = x x 8 x(x+) = x 8 x(x+) 17. Simplify the following, write exact expressions, not decimals from a calculator. Assume the variables represent positive numbers. (a) 90 = 9 10 = 10 (b) 18 = 6 = 8 (c) 8x y 6 = xy (d) x 5 y 8 z 15 = 8x y 6 z 15 x y = xy z 5 x y (e) (16) = ( 16) = () = Write the following radical expressions with a fractional exponent. (a) (b) 5 m = m x = x Write the following in radical form, where appropriate without a negative exponent. (a) (8) = ( 8) = () = (b) (m) 5 = ( 5 m) or 5 (m) (c) (a ) = a 9 = 1 a 9 = 1 a 9 = 1 a a 0. The quadratic formula can be used to solve any quadratic equation in standard form: (a) Write the general quadratic equation in standard form: ax + bx + c = 0 (b) State the quadratic formula that allows you to solve any quadratic equation for the unknown variable. x = b± b ac a 1. Solve each equation for the unknown variable, you may use any appropriate solution technique, but you must show all work. (a) 5x + 5x = 0 5x(x + 9) = 0 5x = 0 x = 0 x + 9 = 0 x = 9 (b) x x = 5 x x 5 = 0 use the quadratic formula x = ( )± ( ) ()( 5) () = ± +0 = ± = ± 11 = 1± 11 (c) m + m = 1 m LCD: m m m + m m = 1 m m + m = 1 m = 9 m = ± (d) a = 6a 5 a + 6a + 5 = 0 (a + 5)(a + 1) = 0 a + 5 = 0 a = 5 a + 1 = 0 a = 1

4 (e) x = x (don t forget to check your solutions) x + 10 = x + 5 x + 10 = (x + 5) x + 10 = x + 10x + 5 x + 8x + 15 = 0 (x + 5)(x + ) = 0 x + 5 = 0 x = 5 Both solutions satisfy the equation. x + = 0 x = (f) x + 5 x = LCD: x x (x) + 5 x (x) = (x) x + 0 = 1x x 1x + 0 = 0 (x 10)(x ) = 0 (g) y y + 9 = 0 x 10 = 0 x = 10 x = 0 x = use the quadratic formula y = ( )± ( ) (1)(9) (1) = ± 9 6 = ± 7 = ±i. Marissa and Mark go for a morning workout along the Cocoa Beach Pier. Mark jogs while Marissa roller-blades. Marissa roller-blades miles per hour faster than Mark jogs. When Marissa has roller-bladed 5.8 miles, Mark has jogged. miles. Find Mark s jogging speed. Let x represent Mark s jogging rate, so x + is the Marissa s roller-blading rate. d r t. Mark. x x 5. Mariss 5.8 x + x+. Times must be equal x = 5.8 x+.x = 5.8x 10. =.x.5 = x Mark jogs at a rate of.5 miles per hour.. For the quadratic function, y = f(x) = x + x + 1 answer the following: (a) opens upward or downward? why? opens downward due to a = 1 (the coefficient attached to x is negative) (b) Find the y-intercept: Set x = 0 y = 1 (0, 1) (c) Find the vertex: the x-coordinate occurs at x = b a = ( 1) = = for the y-coordinate substitute x = and find y y = () + () + 1 = = 16 Vertex is at the ordered pair: (, 16) (d) Find the x-intercept(s) (if they exist): x-intercept(s) must exist due the orientation of the parabola, set y = 0 x + x + 1 = 0 x x 1 = 0 (x 6)(x + ) = 0 x 6 = 0 x + = 0 x = 6 x = x-intercepts occur at (6, 0) and (, 0).

5 (e) Draw the graph base on the information in (a-d):. A quarter is thrown from the top of a building in downtown Seattle. The height h, in feet above the ground after t seconds is given by the equation, h = 16t + 6t (a) From what height is the quarter initially thrown? Set t = 0 which shows that h = 19 or 19 feet high. (b) What is the height after the first second of travel? Set t = 1 which shows that h = = 0 or 0 feet high (c) After how many seconds will it hit the ground? set h = 0 and find t 16t + 6t + 19 = 0 t t 1 = 0 (t 6)(t + ) = 0 t 6 = 0 t = 6 t + = 0 t = Obviously, the quarter hits the ground after 6 seconds. (d) Find the time it takes the quarter to reach its maximum height, as well as the maximum height the quarter reaches. (hint: think vertex) Here we are finding the vertex. Maximum time occurs at t = b a = 6 ( 16) = 6 = substitute for t and solve for h the maximum height. h = 16() + 8() + 19 = 56 Reaches a maximum height of 56 feet in seconds. 5

6 5. Sketch the graphs of the following on the same axis. Plot at least ordered pairs and label each graph. y = x and y = log (x) 6. (a) Write the following in exponential form: log 5 (65) = 5 = 65 (b) Write the following in logarithmic form: ( 1 ) = 16 log 1 (16) = 7. Calculate the following logarithms: a. log (6) = x x = 6 but 6 = 6 b. log 5 ( 5) = x 5 x = 5 write radical term with fracitonal exponent. x = 6 x = 6 5 x = 5 1 x = 1 8. Solve for the unknown variable: a. log m (5) = b. log (x + 5) = m = 5 m = 5 = x = x + 5 x = Use a calculator to find each value of y rounded to four decimal places. a. log y =.8 b. lny = = y y = e. y.0016 y