Section 4.5 Function Families
Subsection 4.5.1 Overview
In Chapter 2 and Chapter 3, we saw that linear functions have very predictable behaviors depending on the values of m and b in the equation,Student Page 4.5.2 Transformation of Functions
1.
Each of 9 volunteers wears a tag with a number on it. The number on the tag represents your assigned value of
2.
Stand on the
3.
Move to the ordered pair
4.
At your teacher's request, transform
(a)
Repeat Student Page Exercise 4.5.2.2 and Student Page Exercise 4.5.2.3. The point to which you move is the ordered pair indicated in the table, as guided by your teacher. Use a different color of string for each graph.
(b)
What are the values of
Move to: | Equation | |||
---|---|---|---|---|
(c)
Explain why the new graph is located where it is. Use the equation and the graph in your explanation. Compare each new graph to
(d)
In the final column, write the equation of the function defined by each ordered pair.
Return the room to its original condition. Use Desmos to explore transformations of the parent quadratic function,
5.
Continue the investigation using Desmos.
(a)
Enter
(b)
Set up sliders for
(c)
Play with
(d)
Play with
(e)
Play with
6.
(a)
What is significant about
(b)
This form of the quadratic equation,
7.
Your teacher will provide you a Class Code to complete the Desmos activity, Card Sort: Parabolas in the Teacher Desmos Quadratic Bundle. Complete the activity. The fourth slide is challenging because it does not include scales for any of the graphs. Describe how you knew how to sort each of the equations to fit the graphs provided. Provide illustrations in your description.
8.
Quadratic functions can be written in the following forms*:
- Standard Form
- Vertex Form
- Factored Form
(*Some quadratic functions cannot be written in this form. Do you know why?)
(a)
What information about a quadratic function is most evident when it is given in:
(i)
Standard Form?
(ii)
Vertex Form?
(iii)
Factored Form?
(b)
What information about a quadratic function is the same for each form of the equation?
Student Page 4.5.3 Angry Birds and Quadratic Functions
1.
The graph in Figure 4.5.3.1 shows an Angry Bird waiting to be launched to hit a pig. The bird must go from the origin
What equation will allow the Angry Bird to take the right path? Write an equation. Show the work you do to find the equation in the space to the right of the picture.
2.
Find both coordinates of the vertex of the quadratic function in Student Page Exercise 4.5.3.1 Write the vertex as an ordered pair. How do you know your solution is correct?
3.
Suppose the Angry Bird is launched from
4.
Find both coordinates of the vertex of the equation in Student Page Exercise 4.5.3.3. How do you know your solution is correct?
5.
Suppose the Angry Bird was launched from
Homework 4.5.4 Homework
1.
Your teacher will provide you a Class Code to complete the Desmos activity, Match My Parabola in the Teacher Desmos Quadratic Bundle. Describe how you were able to find equations for the points and graphs provided. What do you know about quadratic functions that helped you complete this activity?
2.
Your teacher will provide you a Class Code to complete the Desmos activity, Marbleslides: Parabolas in the Teacher Desmos Quadratic Bundle. Complete the activity. This fun activity gives you a chance to test your ability to transform functions and earn stars.
3.
Complete the student page, Tiling Tables, to extend your newly acquired skills with transformations of functions.
4.
Consider the picture of water flowing from a spout in Figure 4.5.4.1. A coordinate grid has been imposed on the picture so that the
(a)
To what function family does this curve belong? How do you know?
(b)
Fit an equation to the graph you chose for the scale given. Determine the values of the parameters,
(c)
What is a sensible domain for the function that models this flow of water? Explain.
(d)
What is a sensible range for the function that models this flow of water? Explain.
(e)
How would the picture change if the water pressure increased? What parameters would change? How would each change? Why do you think so?
(f)
If the water pressure decreased, how would the picture change? What parameters would change? How would each change? Why do you think so?
(g)
Suppose
5.
(a)
Revisit Exercise 4.2.4.2 and Exercise 4.2.4.4 from Homework 4.2.4. Find the vertex form for each of the quadratic equations in both tables. Record them next to the tables in Exercise 4.2.4.2 and Exercise 4.2.4.4.
(b)
How can you verify that each form represents the same quadratic function?
(c)
For the forms of the same function you entered into the tables in Exercise 4.2.4.2 and Exercise 4.2.4.4, verify that each form of the equation represents the same quadratic function.
(d)
Revisit Exercise 4.2.4.8 from Homework 4.2.4. Determine equations in vertex form for all three graphs. Explain your work. Use grid points to determine equations; do not estimate coordinates of any points.
Student Page 4.5.5 Tiling Tables
1.
Kelly makes square tables and then tiles the tops. To cover the tabletops, she uses quarter tiles for each corner, half tiles along the sides, and full square tiles to fill in the rest. The first 5 sizes in her line of tabletops are shown in Figure 4.5.5.1. Both side lengths of the first tabletop are 10 inches.
(a)
Complete Table 4.5.5.2 for the first 6 tabletops.
(b)
Look for patterns in Table 4.5.5.2. Describe at least 3 patterns you see in the data.
2.
Let the table number,
(a)
Columns A and B:
(b)
Columns A and C:
(c)
Columns A and D:
(d)
Columns A and E:
3.
Find equations for each set of data in Student Page Exercise 4.5.5.2. Enter the equations into the last row of Table 4.5.5.2.
4.
(a)
Write any quadratic equations in standard form.
(b)
Is it possible to factor the quadratic equation? Why or why not?
(c)
Subtract the constant from the standard form of the equation. Write this new equation.
(d)
Compare the graph of the equation in Student Page Exercise 4.5.5.4 to the graph of the equation in Task 4.5.5.4.c. How do these graphs comapre with each other? How do you know?
(e)
Factor the equation you found in Task 4.5.5.4.c.
(f)
Find the vertex of the equation in Task 4.5.5.4.c from the factored form of the equation (in Task 4.5.5.4.e).
(g)
How does this vertex relate to the vertex of the equation in Student Page Exercise 4.5.5.4. Find the vertex of the equation in Student Page Exercise 4.5.5.4
(h)
Write the vertex form of the quadratic equation in Student Page Exercise 4.5.5.4.
(i)
Show that the equations in Student Page Exercise 4.5.5.4 and Task 4.5.5.4.h are equivalent.