Math 152
Study Guide for Exam 4
Instructor: G. Rodriguez
Fall 2013
You may use a 3" × 5" note card (both sides) and a scientific calculator. You are expected
to know (or have written on your note card) any formulas you may need. Think about any
formulas you needed for homework.
This is the list of things you must know how to do for Exam 4:
1. Solve a quadratic equation by using the Square Root Property. Simplify the answer
as much as possible. Simplify radicals and/or rationalize denominators. Express
imaginary solutions in the form a + bi. 8.1
a) 2x ! 3 = 2
2
b) (x ! 5)2 = 32
c)
(x + 3)2 = !25
2. Solve a quadratic equation by using the Quadratic Formula. Simplify the answer as
much as possible. Simplify radicals. Express imaginary solutions in the form a+bi. 8.2
a) x + 8x = !5
2
b) 2x + 5 = 6x
2
3. Solve an application involving a quadratic function (but is NOT a max/min
problem). Remember: for these you are solving a quadratic equation to answer the
question. 8.1/8.2
a) A baseball player hits a pop fly into the air. The function h(t) = !16t 2 + 64t + 5
describes the ball’s height above the ground, h(t), in feet, t seconds after it is hit.
After how many seconds does the baseball hit the ground? Round to the nearest
tenth.
b) The length of a rectangle is 8 feet longer than the width. If the area is 400 square
feet, find the rectangle’s dimensions. Round to the nearest tenth.
c) A rectangle park is 8 miles long and 5 miles wide. How long is a pedestrian trail
that runs diagonally across the park? Round your answer to the nearest tenth.
4. Solve a maximum/minimum application problem. Remember: for these you are getting
the info you need from the vertex. 8.3
a) A baseball player hits a pop fly into the air. The function h(t) = !16t 2 + 64t + 5
describes the ball’s height above the ground, h(t), in feet, t seconds after it is hit.
When does the baseball reach its maximum height? What is that maximum
height?
b) You have 800 feet of fencing to enclose a rectangular plot that borders a river. If
you do not fence the side along the river, find the length and the width of the plot
that will maximize the area. What is that maximum area?
5. Graph a quadratic function of the form f ( x) = a ( x ! h) 2 + k . Label the vertex and
intercepts. The graph must include at least five points. Some points may be obtained
using symmetry. 8.3
a) f (x) = (x ! 3)2 ! 4
b) f (x) = (x ! 3)2 + 4
6. Graph a quadratic function of the form f ( x) = ax 2 + bx + c . Label the vertex and
intercepts. The graph must include at least five points. Some points may be obtained
using symmetry. 8.3
a)
f ( x ) = !x 2 + 6x ! 5
b)
f ( x ) = x 2 ! 4x + 4
7. Solve equations that are quadratic in form by making an appropriate substitution. (You
can also solve these ‘as-is’. See class notes.) Simplify answers when possible. If at
any point in the solution process both sides of an equation are raised to an even
power, a check is required. Remember: in the process of solving these you ended up
having to solve a linear, radical, rational, or quadratic equation. 8.4
a) x ! 13x + 36 = 0
4
2
2
1
b) x 3 ! 9x 3 = !8
3x !2 + 2x !1 = 1
c)
d) x ! 2 x ! 15 = 0
8. Solve a polynomial inequality. Graph the solution set and write in either interval
notation. 8.5
2x 2 + 9x ! "4
9. Given the functions f(x) and g(x) find (g  f)(x), (g  f)(#), (f  g)(x), or (f  g)(#)
Let f(x)=x2+6 and g(x)= 5x–3. Find the following:
a) (g  f)(x)
b) (f  g)(x)
c) (f  g)(4)
9.2
10. Find the inverse of a one-to-one function. Use appropriate function notation for the
answer, that is, f !1 (x) . 9.2
a)
f (x) =
b)
f (x) = (x ! 4)3 + 1
3
x!2 +5
11. Given a logarithmic function, find its domain. Write the answer in interval notation. 9.3
f (x) = log 4 (x ! 3)
12. Evaluate a logarithmic expression without the use of a calculator. 9.3
a) log 4 64
b) log 49 7
c) log 3 811
d) ln e
3x
13. Solve exponential equations. There were two types with different instructions.
9.5
Solve exponential equations by expressing each side as a power of the same base
and then equating exponents.
a) 4 x!2 =
1
64
b) 125 = 25
Solve exponential equations by taking the logarithm on both sides. Express the
solution set in term of logarithms (that is give an exact answer). Then use a
calculator to obtain a decimal approximation, correct to two decimal places.
x
c) 5e
3x
= 400
14. Solve logarithmic equations. Be sure to eject any value of x that is not in the domain of
the original logarithmic expressions. Give an exact answer(s) and then if necessary,
give a decimal approximation correct to two decimal places, for the solution. 9.5
a) log(x ! 5) = 3
b) log 2 (x + 3) + log 2 (x ! 3) = 4
c) log 3 (x ! 1) ! log 3 (x + 2) = 2
d) ln(x + 4) ! ln(x + 1) = ln x
15. Solve applications involving exponential or logarithmic functions. Read the instructions
carefully and round as indicated. You will be given the function and will need to know
how to use it to answer the question asked. For some problems (9.1, 9.3) you were
evaluating the function for a certain #, that is finding f(#). For other problems (9.5)
you were finding when f(x) = #.
9.1/9.3/9.5
a) Students in a psychology class took a final exam. As part of an experiment to
see how much of the course content they remembered over time, they took
equivalent forms of the exam in monthly intervals thereafter. The average
score, f(t), for the group after t months is modeled by the function
f(t) = 80 — 15log(t + 1), where 0 ≤ t ≤ 12.
i. What was the average score when the exam was first given?
ii. When can we expect the average score to be 65?
b) The function P(t) = 82.4e!0.002t models the population of Germany, P(t), in
millions, t years after 2006.
i. What was the population of Germany in 2006? in 2010?
ii. In which year will the population of Germany by 80 million? Give an exact
answer (that is with log’s) and then round to the nearest whole number.