NAME:
Final Exam
MATH 141.09
12 / 9 / 2014
1. If f(x) = 4 − x2 when x < 1
and = −0. 5x + 3.5 when x ≥ 1,
(a) What are the x and y intercepts of the graph of f(x)?
(b) Sketch the graph of f(x).
2. Consider f(x) = x2 − 2x + 2. If you write f(x+h) in the form Ax2 + Bx + C,
(a) What is B?
(b) What is C?
3. Start with the function f(x) = x3 + 3. If you shift the function 2 units to the left and then
shift the resulting function 5 units down and then reflect the resulting function about the
x-axis, what is the final form of the function?
4. The vertex of a triangle is given by a point (x,y) on the graph of the function f(x) = 7 − x2.
Another vertex is at the origin and the third vertex is on the x-axis having the same
x-coordinate as the point (x,y). What is the area of the triangle as a function of x?
(Hint: See figure on blackboard)
5. Solve the equation | (x/3) − (3/4) | = 2.
6. What is the center and radius of the circle given by the equation:
x2 + y2 − 6x + 4y + 9 = 0
7. Given two points P1 = (−3, −5) and P2 = (5, 1),
(a) What is the distance from P1 to P2?
(b) What are the coordinates of a point that is 3 times as far from P1 as P2?
8. What is the equation of the line (in the form y = mx + b) passing through points P1 and P2 in
the previous problem?
9. (a) Write the function f(x) = 3x2 + 6x − 8 in the form a(x − h)2 + k.
(b) Sketch the graph of f(x), specifying the vertex and all intercepts in the form (x,y).
10. (a) What are the possible rational zeroes of the function f(x) = 2x4 + 3x3 − x2 − 4 ?
(b) Which of these possible rational zeroes are actual zeroes of f(x) ?
(Hint: Use synthetic division, remembering that the coefficient of the x term in f(x) is zero)
11. (a) Sketch the graph of the function f(x) = (3x2 + 8) / (x2 − 4) .
(b) What is the domain and range for f(x)?
12. One root of f(x) = x3 − 3x2 + 4x − 2 is 1 − i. Find the other two roots.
13. (a) What is the zero of f(x) = −1 + log10(x − 1) ?
(b) Sketch the graph of f(x) .
14. Solve the equation: 7x+2 = 14x .
15. What is log723 rounded to 3 decimal places?
16. What annual interest rate would be required if you wanted to double your money in 10
years if the interest is compounded daily?
17. If f(x) = 1 / (x + 3) and g(x) = 3x2 − 2,
(a) What is (f◦g)(3) ?
(b) What is (g◦f)(−6) ?
18. If you wanted the sector of a circle to have the same area as a square with the length of its
side equal to the radius of the circle, what would the central angle of the sector have to be in
radians? (Hint: See figure on blackboard)
19. If a race car is going 100 miles per hour around a circular track having a quarter mile
radius,
(a) What is the angular speed of the race car in radians per hour?
(b) How many minutes does it take the race car to go once around the track?
20. In a right triangle, if the acute angles are α and β and tan(α) = 1/3, what is cos(β) ?