### Student Page 4.2.3 Multiplying Lines Electronically With Desmos

Now that you have graphed the product of two lines by hand, explore the product of two lines using Desmos as you complete the student page, Multiplying Lines Electronically With Desmos. Do you get the same results? What more do you notice?

#### 1.

Set up Desmos to play with products of linear functions, as follows:

##### (a)

Type \(y = x - p\) into the first input line. Click on the \(p\) button to create a slider for \(p\text{.}\) Start the slider at \(p = 0\text{.}\)

##### (b)

Type \(y = x - q\) into the first input line. Click on the \(q\) button to create a slider for \(q\text{.}\) Start the slider at \(p = 1\text{.}\)

##### (c)

Type \(y = (x - p)(x - q)\) into the third input line.

##### (d)

Type \(y = x^2\) into the fourth input line.

#### 2.

Play with the \(p\) slider.

##### (a)

How does the graph of the product of two lines change as you change \(p\text{?}\)

##### (b)

Why does the change in \(p\) cause the product graph to appear as it does?

##### (c)

What happens to the product graph with \(p\) is less than 0? Why does that make sense?

##### (d)

From your previous mathematical experiences, what is the graph of the product called?

#### 3.

Play with the \(q\) slider until you can explain what is happening to the product graph and why it is happening. We call the graph to the product of two lines a parabola. A parabola is the graph of the quadratic function.

#### 4.

Edit the first input line to be \(y = a (x - p)\text{.}\) Create a slider for \(a\text{.}\) Edit the third input line to be \(y = a (x - p)(x - q)\text{.}\) Play with the \(a\) slider.

##### (a)

For what values of \(a\) does the parabola appear to be wider than \(y = x^2\text{?}\) Why do you think \(a\) has this effect on the parabola?

##### (b)

For Task 4.2.3.4.a, is the parabola wider than \(y = x^2\text{?}\) Why or why not? Argue from the equations, not the appearance of the graphs.

##### (c)

For what values of \(a\) does the parabola appear to be steeper than \(y = x^2\text{?}\)

##### (d)

Explain what \(a\) does to the parabola.

#### 5.

Turn off \(y = x^2\) (click on the colored button to the left of the equation). Reset the \(a\) slider so that \(a = 1\text{.}\) Play with the sliders for \(p\) and \(q\text{.}\) Compare the lines to the parabola. What do the lines and teh parabola always have in common no matter how you change \(p\) and \(q\text{?}\)

#### 6.

Continute to play with all three sliders, \(a\text{,}\) \(p\text{,}\) and \(q\text{.}\)

##### (a)

Try each of these three possibilities:

###### (i)

Both lines have positive slopes.

###### (ii)

Both lines have negative slopes. (What additional change do you need to make?)

###### (iii)

One line has a positive slope and the other line has a negative slope.

##### (b)

How can you tell from the lines what a parabola created from the product of the lines will open upward? Reason from the equations of the lines and the parabola.

##### (c)

How can you tell from the lines what a parabola created from the product of the lines will open downward? Reason from the equations of the lines and the parabola.

##### (d)

How can you tell from the lines where the parabola will intersect the \(x\)-axis?

##### (e)

How can you tell from the lines where the parabola will intersect the \(y\)-axis?

##### (f)

What is a line of symmetry?

##### (g)

How can you find the line of symmetry for a function that is the product of two lines? Why does this make sense?

##### (h)

Each parabola has a highest or lowest point, called the vertex. What information about the vertex can you determine from the lines? Explain.