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.
Use factoring by grouping and then recognize the difference of two squares:
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:
Then solve both inequalities:
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: