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Math 112 – College Algebra I
EXAM FOUR: Sections R.8, 3.1 and 3.
Tuesday, April 21, 2009
Name: Answer Key – Version B
Instructor: ClassTime/Section:
- Show all work to receive credit for each of the problems.
- Incorrect answers with incorrect work shown or no work shown will NOT receive any credit.
- Circle your answers and when appropriate label them.
- Give answers to written questions in complete sentences.
- (3 points each) Simplify the expression. Assume that all variables are positive. Write your answers in radical notation. a. € 27 x 5 y 3 3 b. € 9 x^5 b^6 (^4) • b 2 9 x^5 4 € 3 xy x^2 3 €
b 4
b c. € (^6) s (^5) p • 4 s (^3) p (^3) d. €
s (^56) p (^16) s (^34) p (^34) s (^1912) p (^1112) = s^12 s^7 p^11 €
3 − 5 3 = 5 5 3 e.
(^ x^ −^2 ) ( x^ −^5 ) =
x − 2 x − 5 x + 10 = x − 7 x + 10
- (4 points each) Let € f (^) ( x ) = (^2) ( x − (^4) ) 2 + 3. a. Identify the vertex and leading coefficient.
Vertex = (4,3) Leading Coefficient = 2
b. Write the function as € f (^) ( x ) = ax 2 − bx + c € f (^) ( x ) = (^2) ( x − (^4) ) ( x − (^4) ) + 3 f (^) ( x ) = 2 x^2 − 4 x − 4 x + 16
f (^) ( x ) = 2 x^2 − 8 x − 8 x + 32 + 3 f (^) ( x ) = 2 x^2 − 16 x + 35
- Let € g (^) ( x ) = x^2 + 4 x − 5. a. (5 points) Use algebra to find the x - intercepts of € g (^) ( x ). € x 2
- 4 x − 5 = 0 (^ x^ −^1 ) ( x^ +^5 ) =^0 x − 1 = 0 or x + 5 = 0 x = 1 or x = − 5 (^1 ,^0 ) (− 5 ,^0 ) b. (5 points) Draw the graph of € g (^) ( x ) = x^2 + 4 x − 5. Make sure to include at least 3 points that fall on the curve. x =
(^2) ( (^1) )
f (^) (− 2 ) = (^) (− 2 ) 2
- (^4) (− 2 ) − 5 = − 9 vertex : (^) (− 2,− (^9) )
- The function € f (^) ( x ) = −0.1352 x 2 + 12.1581 x − 22.7399 models the distance, in feet, a baseball travels depending on the angle, x , at which the ball is hit, for a bat speed of 90 miles per hour. a. (2 points) Use this quadratic model to determine the angle in degrees at which the ball must be hit to achieve the maximum distance. Give answer to the nearest degree. (You can solve this graphically or symbolically.)
This is the x coordinate of the vertex.
x = −( − 12.1581) (^2) (− .1352)
b. (2 points) Use this quadratic model to determine the maximum distance the ball can be hit. Give answer to the nearest foot. (You can solve this graphically or symbolically.)
This is the y coordinate of the vertex.
f (^) ( (^45) ) = −0.1352 (^) ( (^45) ) 2
- 12.1581( (^45) ) − 22.7399 = 250.59 251 feet
- (2 pts each) For each statement below circle T if the statement is true and F if the statement is false. T F The vertex of the graph of € f (^) ( x ) = − (^4) ( x + (^1) ) 2 + 3 is € (^ − 1 ,^3 ). T F The minumum of the graph of € f (^) ( x ) = − (^4) ( x + (^1) ) 2 + 3 is attained at the vertex. T F €
3
- 5 = 3 5 T F The expression € − (^) (− 4 ) ± (^) (− 4 ) 2 − (^4) ( (^1) ) ( (^3) ) (^2) ( (^1) ) can be used to find the solutions to the quadratic equation € x^2 − 4 x + 3 = 0. T F The x - intercepts of the graph of € f (^) ( x ) = (^) ( x − (^5) ) ( x + (^7) ) are € (^5 ,^0 ) and € (^ − 7 ,^0 ).
- (2 points each) Use the graph of the quadratic function, € f (^) ( x ) at the right to answer the following questions. a. Determine the domain of the function. € (^ −∞ ,∞) All Real Numbers b. Determine the range of the function. € [^ − 9 ,∞) or € y ≥ − 9 c. Determine the vertex of the function. € (^1 ,−^9 ) d. Determine the equation of the axis of symmetry. € x = 1 e. Find all values of x such that € f (^) ( x ) = 0. € x = −
= −.5 x =
f. Circle the best response. The sign of the leading coefficient is: I. positive II. negative III. It cannot be determined from the graph. g. Determine the y - intercept(s) of the function.
y=-5 or (0,-5)
h. Find € f (^) (− 1 ). € f (^) (− 1 ) = 7 i. Find the average rate of change of f (^) ( x ) from € x 1 = − 1 to € x 2 = 0. m = 7 − (^) (− 5 ) − 1 − 0
