Math 1150: Precalculus, Instructor: Dana Ernst
Solutions for Midterm Exam #1
Here are partial solutions to your first midterm exam. It would be in your best interest to take the time to figure out where you made mistakes. As always, if you have questions, please ask.
- Factor completely (over the integers) each of the following. (6 pts each)
(a)
Use factoring by grouping and then recognize the difference of two squares:
(b)
First factor out an
and then factor the remaining quadratic:
2. Solve the following inequality. Write your answer using interval notation. (8 pts)
First, isolate the absolute value:
Next, set up two inequalities:
and
Then solve both inequalities:
and
Using this information, we can write the solution using interval notation:
3.
Find the standard equation for the circle that is centered at
and passes through the point
. (8 pts)
Recall the standard equation for a circle:
Since the circle is
centered at
,
we know h and k. To
find r, use the fact
that the
circle passes through the point
:
So, the equation of the circle is:
4.
Find the equation of the line that passes through the points
and
. Write your answer in slope-intercept
form. (8 pts)
First, find the slope:
Then plug either given point into the point-slope form of a line:
Lastly, solve for y:
5.
Find the equation of the line that passes through the point
and is perpendicular to
. Write your answer in slope-intercept
form. (8 pts)
First, find the slope of the given line by solving for y:
This implies that the
slope of the
perpendicular line is
(the negative reciprocal). Then
plug this slope and the given
point into the point-slope formula and solve for y:
6. Give an example of an equation using the variables x and y that does not represent y as a function of x and explain why. (6 pts)
Anything where y has a nonzero even power will do. For example:
If you solve for y, you will get two functions. So, the original equation cannot be a function.
7. The population of nuggets has been drastically decreasing for the past few years. In the year 2000, there were 10,000 nuggets living in Nuggetville. This year (2005), there are only 1000 nuggets remaining. What is the average rate of change in the population of nuggets over this time period? Label your answer with appropriate units. (8 pts)
Let
represent the population at time t. Then
the average rate of change in the population over the five year time
period is:
8.
Find the difference quotient of the function
. Simplify your answer
completely. (8 pts)
Here, you just need to remember the formula and then simplify by getting common denominators and cancelling:
9.
Sketch the graph of
. Label at least 2 points on the graph
with their corresponding x
and y-coordinates.
(6 pts)
f is the graph of
reflected down, squished by a half,
shifted
left 2, and down 2. One of the
points that you can label is the vertex, which is
. Another easy point to label is the y-intercept, which is
(just plug in
).
10.
Let
and
. (6 pts each)
(a)
Find
. Simplify your answer.
We see that:
(b)
What is the domain of
?
Recall that the domain
of
is the intersection of the domains of f and g,
except we must also exclude any values that make g equal to 0 (can’t divide by 0). The
domain of f is
and the domain of g is
. There are no x-values that make g equal to 0.
So, the domain of
is
.
11.
Let
. Find functions f, g, h such
that
. (8 pts)
Recall that function composition works from the inside out (right to left). So, we want to h to be the first thing that k does to an input, g to be the second thing that k does, and f to be the third thing that k does. So, we get:
12.
Does the function
have an inverse? If yes, then
find it. If no, then explain why. (8 pts)
This function does have an inverse. To find it, switch x and y and then solve to y: