### Student Page2.7.2Linear Functions with Desmos — Slopes and y-Intercepts A4US

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.