Student Page 2.7.2 Linear Functions with Desmos — Slopes and y-Intercepts
Use Desmos 33 to complete the following investigation. Be ready to discuss your findings with your group and with the class.
1.
Type \(y = mx + b\) into the entry line on the left side of the screen. Notice that Desmos allows you to create sliders for \(m\) and \(b\text{.}\) Do this by clicking the “all” button.
2.
Set \(b = 0\text{.}\) Play with \(m\text{.}\) Describe the appearance of the line when:
(a)
\(m = 0\)
(b)
\(m \lt 0\)
(c)
\(m \gt 0\)
(d)
What effect does changing \(m\) have on the graph of \(y = mx\text{?}\)
(e)
Why does \(m\) have the effect you say it does?
3.
Graph the line \(y = x\) by typing it into the next entry line on the left side of the screen.
4.
(a)
Which line is steeper, one with a slope of 2 or one with a slope of 0.25? Why?
(b)
When \(b = 0\text{,}\) how do each of these lines compare with \(y = x\text{?}\) Why is this sensible?
5.
(a)
Which line is steeper, one with a slope of \(-2\) or one with a slope of 0.25? Why?
(b)
When \(b = 0\text{,}\) how do each of these lines compare with \(y = x\text{?}\) Why is this sensible?
6.
Let \(b = 0\text{.}\)
(a)
What is the equation of the line when \(m = 2\text{?}\)
(b)
What is the equation of the line when \(m = 0.25\text{?}\)
(c)
What is the equation of the line when \(m = -2\text{?}\)
7.
Set \(m = 1\text{.}\) Play with \(b\text{.}\) Describe the appearance of the line when:
(a)
\(b = 0\)
(b)
\(b \lt 0\)
(c)
\(b \gt 0\)
(d)
What effect does changing \(b\) have on the graph of \(y = mx + b\text{?}\)
(e)
Which direction is the graph moving, up and down or left and right? How do you know?
8.
(a)
In your own words, what is slope?
(b)
State everything you can about the value of m and the appearance of the graph of \(y = mx + b\text{.}\)
(c)
In your own words, what is the \(y\)-intercept?
(d)
State everything you can about the value of \(b\) and the appearance of the graph of \(y = mx + b\text{.}\)
9.
Predict the appearance of the graph of each of the following lines. Do not draw them yet.
(a)
\(y = 0.5x + 1\)
(b)
\(y = -x - 2\)
(c)
\(y = 3x - 0.5\)
(d)
Hold an empty page protector against your computer/tablet screen. Use the viewing window on your screen and hand-draw each graph using a dry erase marker. (Trace the \(x\)- and \(y\)-axes on the page protector so you can accurately reposition the graphs if the page protector slides.)
(e)
Without changing the viewing window, electronically graph each of the equations in Task 2.7.2.9.a–2.7.2.9.c. Compare the Desmos graphs with your hand-drawn graphs. If any graph is incorrect, decide what caused the error.
10.
Summarize your work. Answer the following questions and any others that occur to you.
(a)
Given a graph, how can you determine its slope?
(b)
In an equation of a line, \(y = mx + b\text{,}\) which letter, \(m\) or \(b\text{,}\) represents the slope? Explain.
(c)
Given a graph, how can you determine its \(y\)-intercept?
(d)
In an equation of a line, \(y = mx + b\text{,}\) which letter, \(m\) or \(b\text{,}\) represents the \(y\)-intercept? Explain.