# Exploring Mathematics with the Inequality Graphing Application

# Exploring Mathematics with the Inequality Graphing Application

## Introduction

A day at an amusement park is full of excitement. Roller
coasters flip, dip, and twirl

you around. Remember when roller coaster thrills were off limits to you because
you

were not yet tall enough to meet the height requirement? Amusement parks

require that a person must be at least 48 inches tall in order to ride roller
coasters.

You can represent “at least” or “at most” situations with
mathematical inequalities.

In this exploration, you will write an inequality using two variables . You will
also

graph two- variable inequalities in a Cartesian coordinate plane .

## Problem

During a basketball game, teams receive one point for free
throws scored, two

points for field goals, and three points for long-range field goals. In last
weekend’s

game, the Shooting Stars scored a total of 60 points from field goals, both
regular

and long-range.

How many of each type of field goal did the Shooting Stars
possibly make? Write a

linear equation that represents this situation.

_____________________________________________________________________________

How would your equation change if the Shooting Stars had
scored more or less than

60 points?

_____________________________________________________________________________

_____________________________________________________________________________

## Exploration

1. Let x represent the number of 2-point field goals
scored by the Shooting Stars

and y represent the number of 3-point field goals scored by the Shooting Stars.

Write an equation that represents the total points scored from field goals in
last

weekend’s game: ___________________________________

2. Rewrite this equation in slope-intercept form (y = mx +
b): ____________________

What is the slope? __________ What is the y-intercept? __________

3. Clear all equations in the Y= editor of the graphing
handheld. Press or
to

highlight the equation, and then press Also,
be sure that all plots are

turned off. Because x and y refer to the number of baskets scored, you will use

integer values for both .

4. Set a viewing window with integer values by

pressing 8 to select 8:ZInteger. Check

that x = 0 and y = 0. (This indicates that the

viewing window will be centered at the

origin.) Press

5. Press and enter the
values shown.

Do these maximum and minimum x and y

values show an appropriate viewing window

for the basketball problem? Explain your

thinking.

__________________________________________

__________________________________________

6. Enter the slope-intercept form of your

equation in the Y= editor of your graphing

handheld. To do so, press At Y= enter

your equation. Press Sketch the graph

that you see on your screen.

7. You can use the TRACE feature on the graphing handheld
to complete a table

of values that represent possible solutions. Press
to access the

FORMAT Menu. Select CoordOn. Press

The coordinates x = 0 and y = 20 should appear on the
screen of your graphing

handheld. What do these values represent with respect to the basketball

problem?

x = ________________________ y = ________________________

These values have been recorded in the table shown.

Next, you will find other values to complete the table.

8. Press to trace to x
= 1. What is the corresponding y-coordinate? _________

Does this ordered pair represent a solution to the problem ? Why or why not?

________________________________________________________________________

________________________________________________________________________

If it does, add this ordered pair to the table of values.

Continue tracing the integer values for x, from x = 2
through x = 30. Record all

the ordered pairs that are possible solutions to the basketball problem in the

table of values above.

The set of all x-coordinates in a function is the domain.
The set of y-coordinates

in a function is the range. Use your completed table of values to list the
domain

and the range of the function that represent the basketball problem.

Domain: ______________________________

Range: ______________________________

From the table, you can see that as the y-values decrease
by 2, the x-values

___________________.

Would you describe the relationship between the x-values
and the y-values as a

constant or variable? What type of function exhibits this characteristic?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

Describe this relationship in terms of the number of
3-point field goals and

the number of 2-point field goals.

________________________________________________________________________

________________________________________________________________________

9. In step 2 you determined that the y-intercept for this
linear relationship is

(0, 20). What is the significance of this point?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

10. Press to trace to
x = 36. What is the corresponding y-value? _____________

Verify algebraically that these coordinate points satisfy the equation from

step 1 by substituting the x - and y-values into the equation.

The ordered pair (36, -4) satisfies the equation in step
1. Can it be a solution to

the basketball problem? Explain why or why not.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

Identify three ordered pairs that satisfy the equation but
are not solutions to

the problem.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

11. In their next game, the Shooting Stars want to score
at least 60 points from field

goals. Rewrite the equation from step 1 as an inequality to show this situation.

________________________________________________________________________

Write this inequality in slope-intercept form:

________________________________________________________________________

12. Use the Inequality Graphing App to graph

your inequality. Press Select Inequalz.

Press and then press any key to

continue. The equation you entered in step 2

is displayed. At the bottom of the screen, you

should see the equal sign and four inequality

signs.

13. Highlight = next to your equation. Press

to replace the = with the “greater

than or equal to” symbol (≥).

14. Press (The Shades
and PoI-Trace

buttons appear. You will learn about these

features in later activities.)

Describe the graph. How does it show that

the Shooting Stars scored at least 60 points

from field goals?

__________________________________________

__________________________________________

15. Use the cursor keys to navigate about the

viewing window. (Press to deactivate

the TRACE feature, if necessary.)

Select a point from the shaded portion of the

graph and record the coordinates:

__________________________________________

Verify algebraically that the point satisfies

the inequality from step 11 by substituting

the x- and y-values into the inequality.

16. Change the inequality in the Y= editor to

2x + 3y > 60. (Be sure to change the

inequality to the slope-intercept form.)

Sketch the graph on the grid. How does the

graph change?

17. Change the inequality to 2x + 3y < 60. Sketch

the graph in the grid. How does the graph

change?

18. Verify your graphs using the Inequality

Graphing App. Enter each inequality in the

Y= editor and press

How closely do your sketches match the

graphs from the Inequality Graphing App?

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