Transformations and functions: vertical shift
The graph of a function y = f(x) is vertically shifted H units up when you add H to the function.
For example, if f(x) = x2 and g(x) = x2 + 1, then the graph of g(x) is the same as the graph of f(x), shifted up one unit.
Rewrite the following functions with the indicated shift. Check the graphs with a graphing program such as desmos.com:
1. Shift up 2 units: f(x) = 2x – 4
2. Shift down 3 units: f(x) = 4x3
3. Shift up 10 units: f(x) = sqrt(x) (f(x) = square root of x).
4. Shift up 5 units: f(x) = 1/x
5. Shift down 100 units: f(x) = x1.5
For each diagram, copy and draw an axis of symmetry. There is at least one axis in each figure.
Example: The square below has a red line where one axis of symmetry is.
The word “extrema” means “maxima and/or minima”. In turn, maxima is the plural of maximum, and minima is the plural of minimum. So this section is about maxima and minima.
Consider the following data about a function f(x).
– the input variable is a human being, i.e. x stands for a human being who lives on planet earth.
– the output of f is the height of the human being, measured in cm.
– the tallest thirteen year old girl in the city of Palo Alto in N. America has a height of 175 cm (about 5’9″).
– the tallest thirteen year old girl in North America has a height of 190 cm (about 6’3″).
– the tallest thirteen year old girl in the Europe/Asia combined continent has a height of 200 cm (about 6’8″).
– the tallest thirteen year old girl in South America is shorter than the tallest thirteen year old girl in Asia.
– the tallest thirteen year old girl in Africa has a height of 199 cm.
– the tallest thirteen year old girl in Australia has a height of 198 cm.
– the tallest thirteen year old girl in Antarctica has a height of 165 cm.
– the tallest thirteen year old girl in islands other than the named continents has a height of 173 cm.
– the shortest thirteen year old girl in Palo Alto has a height of 123 cm.
– the shortest thirteen year old girl in North America has a height of 123 cm.
1. What is the full domain of f? (Do you know approximately how big this domain is?)
2. Is f(x) a bounded function? If so, what are the bounds of f(x)?
3. From the above data, give an example of a local maximum of f(x). Specify what the value of the local maximum is, and where it occurs, as best you can.
4. In your specific example of a local maximum of f(x) in problem 3, what is the subdomain where the local maximum occurs?
5. In the above data, if the domain is restricted to thirteen year old girls, does f(x) have an absolute maximum?
Average Growth Rates
The average growth rate of a function f(x) is the average rate of increase (or decrease) over an interval of the x-axis. Here is how to calculate the average growth rate:
The x-axis interval can be described as [a, b]. The average growth rate is:
( f(b) – f(a) ) / (b – a)
Example: if f(x) = x2, the average growth rate of f(x) on the interval [2, 4] is:
( 42 – 22 ) / (4 – 2) = (16 – 4) / (4 – 2) = 12/2 = 6
You have probably heard of average growth rates before. If f(x) = mx + b, the average growth rate is the slope. If you have studied distance-time problems, the average growth rate is the average speed.
For each of the functions below, calculate the average growth rates on two intervals: [0, 1] and [1, 2]. First think: what do you expect the answers to be. Do the answers fit your expectations?
1. f(x) = 2x + 5
2. f(x) = |x – 1|
3. f(x) = x – 1
4. f(x) = sqrt(x) (the square root of x). For this problem, take the square root of 2 to be 1.414.
5. f(x) = -(x – 1)2
Function values can increase, decrease, stay the same or neither. Sometimes the behavior differs depending on the domain. For each function below state whether it is increasing, decreasing, or staying the same on the specified domain. (Hint: the functions below do not wiggle around a lot. In the specified domain they are either increasing, decreasing or staying the same).
1. f(x) = 5x – 6 domain: all x
2. f(x) = 19 domain: all x
3. f(x) = -3x + 75 domain: all x
4. f(x) = x2 domain: x is > 0.
5. f(x) = 1/x domain: x is not equal to 0.
If a function’s range has some limits, the function is bounded. If the function’s range has no limits, the function is unbounded. Are these functions bounded or unbounded? If bounded, do they have an upper bound, lower bound, or both?
1. f(x) = 3x -9
2. f(x) = -x2
3. f(x) = |x – 2|
4. f(x) = x3
5. f(x) = 1/(x2 + 1)
Domain and Range
What are the domain and range of each function?
1. f(x) = |x|
2. f(x) = 2x – 3
3. f(x) = x – 4
4. f(x) = 1 – |x|
5. f(x) = (1 – |x|)2
Maxima and Minima
For each of these absolute value functions: is there an absolute maximum, minimum, or neither? If there is a minimum or maximum, specify the x value where it is located.
1. f(x) = |x| + 1
2. f(x) = |x – 1|
3. f(x) = 1 – |x|
4. f(x) = 1 – |2x|
5. f(x) = |x| + |x – 1|
Bonus question: graph the function in #5.