Problem Set for Previewing Unit 1
In the first unit of math I the students are taking the ideas they learned before involving functions. The students investigate different “parent functions” and the effects that changing parameters will have on the graph. The students will investigate the following parent functions: Constant, Linear, Quadratic, and Cubic. In addition to using function notation, students will be required to understand and state the characteristics of the functions. These characteristics involve domain, range, x and y intercepts Maximum or Minimum, Interval of Increasing, Interval of Decreasing, and End Behavior.
Vocabulary
Function Domain Range X-Intercepts Y-Intercepts Maximum Minimum Interval Infinity Transformation
Scenario
In Queenstown New Zealand there exists what is known as the Ledge; a runway platform 400 meters (≈1312 ft or 130 storeys) above the city. The purpose of this platform is simple, to take a leap of faith like none other in the world. The Ledge provides thrill seekers an opportunity to see what it feels like to jump and fall from 400 meters in the air, luckily there is a bungee cord to prevent any messy clean up. Bungee Jumping is an extreme sport which over the years has garnered a lot of attention. Using small scale examples of a Bungee we will get to investigate functions.
Investigation
Check out the M&M Slinky Investigation
Some possible modifications:
- Have the slinky’s hung from the ceiling prior to the students coming into class
- Instead of 1 m&m at a time I would go with a faster pace of 2 or 3 m&m’s; it would be better to have heavier items (example dice, poker chips, etc).
Additional
Questions
Imagine if we want to move our bungee an additional 15 cm up. Graph a new scatter plot and find the equation of this new line. How did this change the equation of the line from our investigation? How did this change the scatter plot? Predict what would happen if instead we moved the bungee 15 cm down. Can you predict what the change in the equation of the line would be? How can we transform the original graph?
Discussion Questions
1) Looking at the scatter plots, we know that the x-axis represents the number of m&m’s inside of the cup. What is the smallest number of m&m’s that one may have in the cup? What is the largest number of m&m’s that will fit in the cup?
a. With these questions we want to bring up the idea of a domain of a function and the concept of infinity. In the cup we have a smallest value of 0, but we can place as many m&m’s in the cup as we want since it is the independent variable, thus the largest number of m&m’s in the cup is infinity.
2) Looking at the scatter plots, we know the y-axis represents the distance from the cup to the floor. What is the greatest distance that the cup may be from the floor? What is the least distance that the cup and the floor may be?
a. With these questions we want to bring up the idea of the range of a function. The greatest distance was the starting point or the y-intercept, while the smallest distance was zero since we cannot go below the ground no matter what the weight of the cup.
3) Where the graph crosses the x and y axis relates a lot of important information, where are the y-intercept and the x-intercept on our scatter plot?
4) Let’s take a look at the parent function f(x) = x. Create an XY chart to graph this function. Use the spread sheet here to see an example.
a. Looking at the graph of f(x) = x, we know the x-axis represents the inputs of the function. What is the smallest number that one may input into this function? What is the greatest number that one may input? What is the domain of the function?
b. Looking at the graph of f(x) = x, we know the y-axis represents the outputs of the function. What is the smallest number that the function output? What is the greatest number that the function output? What is the range of the function?
c. What are the x –intercepts and the y – intercepts of the function? What are these points telling us?
d. Does this graph have a maximum or a minimum? (Recall the definition of maximum and minimum of a graph) Why or why not? Can we say that the maximum or minimum is infinity?
e. From what point to what point is the graph increasing? From what point to what point is the graph decreasing? [Remember that we read graphs from left to right]
f. The view that you see is one snapshot of the graph, recall that the ends of the graph continue beyond what we can see on our paper. As x is increasing what is happening to y? As x decreases what is happening to y? This is called the end behavior of the graph.
5) Repeat number 4 with the parent function f(x) = x2? f(x) = x3? What do you see are the big differences between the three graphs? What do you see that they have in common?
The ability for students to identify parent functions and be able to state the characteristics of a function is very important of the first unit in math 1. The second part of unit 1 deals with the transformations of the parent functions; the additional questions of the investigation helps students see that the transformation of parent functions. Using the attached spreadsheets, we can demonstrate to students these transformations:
Given function f(x) = ax + k
- If a is positive then the graph has a positive slope.
- If a is negative then the graph has a negative slope or is reflected with respect to the x-axis.
- If |a| > 1 then the value of x is multiplied by the parameter a, thus the transformation is described as a “stretch”
- If 0 < |a| < 1 then the value of x is divided by the parameter a, thus the transformation is described as a “shrink”
- The number k represents the movement vertically of the graph. A positive k moves the graph up while a negative k moves the graph down.
With these in mind the students should be able given any function written in the form of f(x) = ax + k describe how one may transform the parent function f(x) = x to get the graph asked for. We may repeat this process with the parent functions f(x) =x2 and f(x) = x3. NOTE: In Math 1 the horizontal shifts are not taught, the horizontal shifts are demonstrated to students in Math 2, thus the h parameter is always considered to be zero.
Given function f(x) = a(x)2 + k
- If a is positive then the parabola is pointing up
- If a is negative then the parabola is pointing down
- If |a| > 1 then the value of x is multiplied by the parameter a, thus the transformation is described as a “stretch.”
- If 0 < |a| < 1 then the value of x is divided by the parameter a, thus the transformation is described as a “shrink.”
- The number k represents the movement vertically of the graph. A positive k moves the graph up while a negative k moves the graph down.
Given function f(x) = a(x)3 + k
- If a is positive then the graph is in the quadrants where both coordinates have the same signs. [Quadrant I and III]
- If a is negative then the graph in the quadrants where both coordinates have different signs. [Quadrant II and IV].
- If |a| > 1 then the value of x is multiplied by the parameter a, thus the transformation is described as a “stretch”
- If 0 < |a| < 1 then the value of x is divided by the parameter a, thus the transformation is described as a “shrink”
- The number k represents the movement vertically of the graph. A positive k moves the graph up while a negative k moves the graph down.
Students should be able to notice that the transformations are the same for each of these parent functions. The parent functions transformations are all very similar students should work on being able to identify all of the transformations of these functions.