### Student Page 2.7.2 Linear Functions with Desmos โ Slopes and y-Intercepts

Use Desmos^{โ33โ} 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 โallโ 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 Taskย 2.7.2.9.aโ2.7.2.9.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.