Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Use the following scenario for questions 1 - 5:
A student has a savings account with $475 in it. She deposits $125 of her paycheck into the account every week. Her goal is to save $7000 for college.

1.

How much will be in the account after 3 weeks?

850

2.

How many weeks will it take until she has $1350?

7

3.

Write an equation that represents the relationship between the dollar amount in her account and the number of weeks of saving.

Compare your answer:

$a = 475 + 125 w$ , where $a$ is the dollar amount in the account and $w$ is the number of weeks of savings.

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4.

Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3 weeks and the week she has $1350. Write down the coordinates. Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above.

Compare your answer:

Line graph showing a steady increase in account amount (dollars) over 55 weeks, starting at $0 and reaching $8,000. The y-axis is labeled amount in account (dollars), x-axis number of weeks.

5.

What is the $x$ -intercept?

Compare your answer:

T $(475, 0)$ .

6.

What other information does the $x$ -intercept identify in the function?

Compare your answer: The $x$ -intercept can identify the zero of the function since it represents where the function equals zero $(y = 0)$ .

7.

What is the the $y$ -intercept?

Compare your answer:

$(0, - 3.8)$ .

8.

What is the slope?

Compare your answer:

$- \frac{1}{125}$ .

Are you ready for more?

Extending Your Thinking

Use the following information to answer questions 1 - 5. A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.

1.

Write an equation that represents the relationship between the gallons of water in the tank and hours the tank has been draining.

Compare your answer:

$a = 450 - 1, 200 h$ where $h$ is time in hours.

2.

Write an equation that represents the relationship between the gallons of water in the tank and seconds the tank has been draining.

Compare your answer:

$a = 450 - \frac{s}{3}$ where $s$ is time in seconds.

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3.

Graph each of your new equations. Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above.

Compare your answer:

Line graph showing the amount of water in a tank decreasing linearly from 500 gallons at 0 hours to 0 gallons at 0.4 hours; x-axis is time (hours), y-axis is amount in tank (gallons).

4.

In what way are all of the graphs the same? In what way are they all different?

Compare your answer:

Your answer may vary, but here are some samples.

They all have the same $y$ -intercept.
They all decrease.
The slopes are different.

5.

How would these graphs change if we used quarts of water instead of gallons? What would stay the same?

Compare your answer:

Your answer may vary, but here are some sample samples: The graphs would still be lines with negative slopes that are different. They would have the same horizontal intercepts. The vertical intercepts would be different, but they would represent the same amount of water.

Self Check

Which of the following points is a solution to the graph below?

$(-2, -6)$
$(7, 2)$
$(0, 3)$
$(2, 7)$

Additional Resources

Writing Equations Using Graphs in Situations

An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables. A solution to such an equation is a pair of numbers that makes the equation true.

Suppose Tyler spends $40 on T-shirts and socks. A T-shirt costs $10 and a pair of socks costs $2.50. If $t$ represents the number of T-shirts and $p$ represents the number of pairs of socks that Tyler buys, what is an equation that represents the equation?

Example 1

Step 1 - Create a two-variable equation.
The cost is $10 per t-shirt $(10 t)$ plus $2.50 per pair of socks $(2.50 p)$ which equals $40.
$10 t + 2.50 p = 40$

Now, we have to graph the equation. We will let $t = x$ and $p = y$ .

Step 2 - Find the $x$ -intercept.
To find the $x$ -intercept, let $p = 0$ .

$10 t + 2.50 p = 40$
$10 t + 2.50 (0) = 40$
$10 t + 0 = 40$
$t = 4$

$(4, 0)$ is the $x$ -intercept.
The $x$ -intercept is also called a solution or zero.

In this scenario, 4 represents the number of T-shirts Tyler can buy if he doesn't purchase any socks with $40.

Step 3 - Find the $y$ -intercept.
To find the $y$ -intercept, let $t = 0$ .

$10 t + 2.50 p = 40$
$10 (0) + 2.50 p = 40$
$(0) + 2.50 p = 40$
$2.50 p = 16$

$(0, 16)$ is the $y$ -intercept.

In this scenario, 16 represents the number of socks Tyler can buy if he doesn't purchase any T-shirts with $40.

Step 4 - Graph the line by connecting the intercepts.

Let’s look at the graph of this equation:

A graph showing a decreasing straight line passing through points (0, 16) and (4, 0), which are highlighted with blue circles and labeled in blue text. The axes are marked with black arrows and numbers.

Let’s reflect about the graph and what it means.

Example 2

What is the slope of the graph?

Solution

$m = - 4$

Example 3

What does the point $(4, 6)$ mean on this graph?

Compare your answer:

If Tyler bought 4 T-shirts and 6 pairs of socks, it would cost more than $40.

Try it

Try It: Writing Equations Using Graphs in Situations

Use Desmos or a graphing calculator to create a graph for $40 x + 20 y = 180$ .

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1.

If $x$ represents the number of pairs of shoes and $y$ represents the number of pairs of jeans, what is one combination that is a solution?

Compare your answer:

Your answer may vary, but here are some samples: $(2, 5)$ , $(1, 7)$ , $(3, 3)$ , $(4, 1)$

2.

What does the combination you identified mean on the graph?

Compare your answer:

Your answer may vary, but here is a sample. For $(2, 5)$ , that means buying 2 pairs of shoes and 5 pairs of jeans will cost $180.

This is how to find a solution and determine its meaning:

Find a point on the graph of the line.
The point $(2, 5)$ is on the graph. Since 2 is the coordinate of the $x$ -coordinate, it represents 2 pairs of shoes.
Since 5 is the $y$ -coordinate, it represents 5 pairs of jeans.
The line represents all combinations that equal $180.

1.5.4 Writing, Graphing, and Solving a Linear Equation

Activity

Are you ready for more?

Extending Your Thinking

Additional Resources

Writing Equations Using Graphs in Situations

Try It: Writing Equations Using Graphs in Situations