Student Page 2.8.2 Big Ideas — Slope and \(y\)-Intercept
Use Desmos on your laptop, tablet, or smart phone. Be ready to share your findings and to demonstrate using the classroom computer.
By now you have discovered that all linear equations can be represented generally by the equation, \(y = mx + b\text{,}\) where \(x\) and \(y\) are variables and \(m\) and \(b\) represent known numbers (\(m\) and \(b\) are called parameters).
1.
Type this general form of a linear equation into Desmos. Create sliders for \(m\) and \(b\text{.}\) For Task 2.8.2.1.a and Task 2.8.2.1.c, use one or two rounds of Roundtable (see Roundtable) to share everything you have learned. Explain each item in your list as you create it.
(a)
Let \(y = mx\) and play with \(m\text{.}\) List everything you can about the effect of \(m\) on the graph of \(y = mx\text{.}\)
(b)
Revisit your list. If you have not already answered the following, do so now. Explain! What happens to the graph of \(y = mx\) when:
(i)
\(m \gt 0\text{?}\)
(ii)
\(m \lt 0\text{?}\)
(iii)
\(\left| m \right|\) is large?
(iv)
\(\left| m \right|\) is close to zero?
(c)
Let \(m = 1\) and play with \(b\text{.}\) List everything you can about the effect of \(b\) on the graph of \(y = mx + b\text{.}\)
(d)
Revisit your list. If you have not already answered the following, do so now. What happens to the graph of \(y = mx + b\) when:
(i)
\(b \gt 0\text{?}\)
(ii)
\(b \lt 0\text{?}\)
(iii)
\(\left| b \right|\) is large?
(iv)
\(\left| b \right|\) is close to zero?
(v)
How is the graph of \(y = mx + b\) moving as you change the value of \(b\text{?}\)
(vi)
Why is the graph moving as you say it does?
2.
(a)
Place a page protector over the screen on your electronic tool. Trace the \(x\)- and \(y\)-axes on the page protector. By hand, graph the following equations. Explain how you drew each graph so that you know it is accurate.
(i)
\(y = 2x + 4\)
(ii)
\(y = 0.5x - 3\)
(iii)
\(y = -x - 1\)
(b)
Once all group members are satisfied with the three hand-drawn graphs from Task 2.8.2.2.a graph them electronically. Was each hand-drawn graph accurate? If not, what needs to be fixed? What were you thinking when you drew each graph? What do you now know? Turn off the electronic graph and redraw the graph using what you now know. Recheck.
3.
Study the graphs. Find an equation for each graph. Do not estimate; use grid points to find each equation accurately. In each case, explain how you know your choices of slope and \(y\)-intercept are correct.
(a)
(b)
(c)
4.
Study the tables. For each table, find an equation to fit the data. In each case, explain how you know your choices of slope and \(y\)-intercept are correct.
(a)
| \(x\) | \(y\) |
|---|---|
| 4 | \(-5\) |
| 8 | \(-7\) |
| 12 | \(-9\) |
| 16 | \(-11\) |
| 20 | \(-13\) |
(b)
| \(x\) | \(y\) |
|---|---|
| 8 | 3 |
| 10 | 11 |
| 12 | 19 |
| 14 | 27 |
| 16 | 35 |
(c)
| \(x\) | \(y\) |
|---|---|
| 1922 | 450 |
| 1928 | 435 |
| 1934 | 420 |
| 1940 | 405 |
| 1946 | 390 |
5.
Study each context. For each context, find an equation. In each case, explain how you know your choices of slope and \(y\)-intercept are correct.
(a)
Fia earns 3 points every time she finds a star in her Dora the Explorer game. She earns a one-time bonus of 50 points in each level for finding a banana for her monkey friend, Boots. Based on the number of stars she finds, how many points will Fia earn in a level if she always finds the banana?
(b)
Valen earns $5 each week for doing his chores. Each time his mom has to remind him, there is a $0.50 deduction from his allowance. How much will Valen earn in a week based on the number of times he needs to be reminded to do his chores?
(c)
Jackson has $18. Each month, he pays $4 to stream music on his iPod. How much does he have or owe based on the number of months he streams music?
6.
In each case below, explain how you find the slope and the \(y\)-intercept in each representation.
(a) \(y\)-intercept, \(b\).
(i)
Table:
(ii)
Graph:
(iii)
Context:
(b) Slope, \(m\).
(i)
Table:
(ii)
Graph:
(iii)
Context: