Question
You can represent a function in different ways: in a table, in a graph,or as an equation.
A day at the amusement park costs $10 for an entrance fee and $2.50for each ride ticket.
TableGraph
Number ofTickets
0
1
2
3
4
Cost ($)
10
12.5
15
17.5
20
Equation in the form ofmath
y
=
m
x
+
b
:
math
y
=
2.5
x
+
10
A graph of cost in dollars versus number of tickets is a line rising from (0, 10) through the points in the table: (1, 12.5), (2, 15), (3, 17.5), and (4, 20).A graph.
image long description
<p>A graph has a curve and a line plotted:</p> <ul> <li>A curve rises from (0, 0) through (1, 1) and (2, 4). The graph of a nonlinear function is not a straight line because there is no constant rate of change.</li> <li>A line falls through (0, 7), (3, 5), and (6, 3). The graph of a linear function is always a straight line because there is a constant rate of change.</li> </ul>
Do You Understand?
What are differentrepresentations of a function?
Use Appropriate ToolsHow can you use agraph to determine that a relationship isNOTa function?
Construct ArgumentsMust the ordered pairsof a function be connected by a straight line ora curve on a graph? Explain.
Do You Know How?
Each week, Darlene tracks the number of partyhats her company has in stock. The table showsthe weekly stock. Is the relationship a linearfunction? Use the graph below to supportyour answer.
A data table and a graph.
image long description
<p>A table of data and a coordinate plane.</p> <p>The coordinate plane has weeks on the horizontal w axis from 0 to 8 in increments of 1 and party hats on the vertical h axis from 0 to 400 in increments of 50.</p> <p>The table of data</p> <table> <tbody> <tr> <td>Weeks, w</td> <td>Party Hats, h</td> </tr> <tr> <td>0</td> <td>300</td> </tr> <tr> <td>1</td> <td>250</td> </tr> <tr> <td>2</td> <td>200</td> </tr> <tr> <td>3</td> <td>150</td> </tr> <tr> <td>4</td> <td>100</td> </tr> <tr> <td>5</td> <td>50</td> </tr> </tbody> </table>~
How can Darlene use the graph above to knowwhen to order more party hats?
A day at the amusement park costs $10 for an entrance fee and $2.50for each ride ticket.
TableGraph
Number ofTickets
0
1
2
3
4
Cost ($)
10
12.5
15
17.5
20
Equation in the form ofmath
y
=
m
x
+
b
:
math
y
=
2.5
x
+
10
A graph of cost in dollars versus number of tickets is a line rising from (0, 10) through the points in the table: (1, 12.5), (2, 15), (3, 17.5), and (4, 20).A graph.
image long description
<p>A graph has a curve and a line plotted:</p> <ul> <li>A curve rises from (0, 0) through (1, 1) and (2, 4). The graph of a nonlinear function is not a straight line because there is no constant rate of change.</li> <li>A line falls through (0, 7), (3, 5), and (6, 3). The graph of a linear function is always a straight line because there is a constant rate of change.</li> </ul>
Do You Understand?
What are differentrepresentations of a function?
Use Appropriate ToolsHow can you use agraph to determine that a relationship isNOTa function?
Construct ArgumentsMust the ordered pairsof a function be connected by a straight line ora curve on a graph? Explain.
Do You Know How?
Each week, Darlene tracks the number of partyhats her company has in stock. The table showsthe weekly stock. Is the relationship a linearfunction? Use the graph below to supportyour answer.
A data table and a graph.
image long description
<p>A table of data and a coordinate plane.</p> <p>The coordinate plane has weeks on the horizontal w axis from 0 to 8 in increments of 1 and party hats on the vertical h axis from 0 to 400 in increments of 50.</p> <p>The table of data</p> <table> <tbody> <tr> <td>Weeks, w</td> <td>Party Hats, h</td> </tr> <tr> <td>0</td> <td>300</td> </tr> <tr> <td>1</td> <td>250</td> </tr> <tr> <td>2</td> <td>200</td> </tr> <tr> <td>3</td> <td>150</td> </tr> <tr> <td>4</td> <td>100</td> </tr> <tr> <td>5</td> <td>50</td> </tr> </tbody> </table>~
How can Darlene use the graph above to knowwhen to order more party hats?
Answers
GPT-5 mini
Short answers to each part:
1) What are different representations of a function?
- Table (list of ordered pairs)
- Graph (points or curve on the coordinate plane)
- Equation/formula (y = f(x))
- Verbal description (words describing the rule)
- Mapping diagram (arrows from inputs to outputs)
2) How can you use a graph to determine that a relationship is NOT a function?
- Use the vertical line test: if any vertical line intersects the graph in more than one point, then a single x-value has more than one y-value, so the relation is not a function. (Example: a circle fails the vertical line test.)
3) Must the ordered pairs of a function be connected by a straight line or a curve on a graph? Explain.
- No. A function is just a set of ordered pairs. If the data are discrete (e.g., counts measured weekly), plot separate points and do not connect them. If the relationship is continuous and described by a linear rule, the points will lie on a straight line; if continuous but nonlinear, they lie on a curve. You only connect points when the context implies continuity.
4) Darlene’s party hats data — is the relationship linear? Use the graph to support your answer.
- Yes. The party-hat count decreases by 50 each week (300, 250, 200, 150, 100, 50). This constant rate of change means the relationship is linear. The equation is h = −50w + 300, and the plotted points lie on a straight line.
5) How can Darlene use the graph to know when to order more party hats?
- Choose a reorder threshold (for example, 50 hats to avoid stockout). Use the graph or equation to find when the inventory reaches that threshold: solve −50w + 300 = threshold → w = (300 − threshold)/50.
- If threshold = 50, w = (300 − 50)/50 = 5 → order at or before week 5.
- If she wants a safety margin (e.g., reorder at 100), w = (300 − 100)/50 = 4 → order at or before week 4.
- In short: read the week where the line crosses your chosen reorder level, and place the order before that week.
1) What are different representations of a function?
- Table (list of ordered pairs)
- Graph (points or curve on the coordinate plane)
- Equation/formula (y = f(x))
- Verbal description (words describing the rule)
- Mapping diagram (arrows from inputs to outputs)
2) How can you use a graph to determine that a relationship is NOT a function?
- Use the vertical line test: if any vertical line intersects the graph in more than one point, then a single x-value has more than one y-value, so the relation is not a function. (Example: a circle fails the vertical line test.)
3) Must the ordered pairs of a function be connected by a straight line or a curve on a graph? Explain.
- No. A function is just a set of ordered pairs. If the data are discrete (e.g., counts measured weekly), plot separate points and do not connect them. If the relationship is continuous and described by a linear rule, the points will lie on a straight line; if continuous but nonlinear, they lie on a curve. You only connect points when the context implies continuity.
4) Darlene’s party hats data — is the relationship linear? Use the graph to support your answer.
- Yes. The party-hat count decreases by 50 each week (300, 250, 200, 150, 100, 50). This constant rate of change means the relationship is linear. The equation is h = −50w + 300, and the plotted points lie on a straight line.
5) How can Darlene use the graph to know when to order more party hats?
- Choose a reorder threshold (for example, 50 hats to avoid stockout). Use the graph or equation to find when the inventory reaches that threshold: solve −50w + 300 = threshold → w = (300 − threshold)/50.
- If threshold = 50, w = (300 − 50)/50 = 5 → order at or before week 5.
- If she wants a safety margin (e.g., reorder at 100), w = (300 − 100)/50 = 4 → order at or before week 4.
- In short: read the week where the line crosses your chosen reorder level, and place the order before that week.