Use the graphs to answer the questions.

Question

(-2,3/4)(1,6) (-2,2/9)(1,6)
A. For each graph, create an input-output table with at least 4 points in each
table.
B. For each graph, write an exponential equation. Explain how you determined
the base and initial value from each graph.
C. Create a real-world situation that can be modeled by each graph.
D. Explain what x, y, a, and b represent in your real-world situations.
E. For each equation, evaluate the equation for a value of x not listed in your
table. Explain what the point represents in the real-world.
F. Create your own unique exponential growth situation, then model it with an
equation, an input-output table, and a graph.
Make the real world situation about a ice cream shop.

Answer 1

Here’s a breakdown of your questions related to the given points and exponential growth, tailored to a situation involving an ice cream shop.

A. Input-Output Tables

Let’s create input-output tables for both sets of points given.

For Graph 1: (-2, 3/4) and (1, 6)

To find additional points, we can interpolate and extrapolate values. However, since specific graphs are not provided, we’ll assume typical behaviors for exponential growth.

| x | y | |----|--------| | -2 | 0.75 | | -1 | 2.25 | | 0 | 3.75 | | 1 | 6 | | 2 | 10.5 |

For Graph 2: (-2, 2/9) and (1, 6)

| x | y | |----|---------| | -2 | 0.222 | | -1 | 0.667 | | 0 | 1.872 | | 1 | 6 | | 2 | 18 |

B. Exponential Equations

Graph 1:

We can assume the equation has the form \( y = a \cdot b^x \).

1. From (1, 6):

   • When \( x = 1\), \( y = 6\) → \( 6 = a \cdot b^1\) → \( 6 = ab\)

2. From (-2, 0.75):

   • When \( x = -2\), \( y = 0.75\) → \( 0.75 = a \cdot b^{-2}\) → \( 0.75 = \frac{a}{b^2}\)

From these two equations:

1. \( a = 6/b \)
2. Substituting into the second equation gives \( 0.75 = \frac{6}{b^3} \implies b^3 = 8 \implies b = 2 \).

Now substituting back gives \( a = 3 \).

So: \( y = 3 \cdot 2^x \)

Graph 2:

1. From (1, 6):

   • \( 6 = a \cdot b\)

2. From (-2, 0.222):

   • \( 0.222 = \frac{a}{b^2}\)

Following similar steps results in \( b = 3 \) and \( a = 18 \).

So: \( y = 18 \cdot 3^x \)

C. Real-World Situations

Graph 1 Situation: An ice cream shop starts with 3 scoops sold on a cold day and doubles sales every warmer day.

Graph 2 Situation: An ice cream cart sells 18 scoops on an ideal sunny day and grows threefold as days progress into summer.

D. Explanation of Variables

• For Graph 1:

  • \( x \): Number of warmer days from the cold day.
  • \( y \): Number of scoops sold.
  • \( a \): Initial scoops sold (3).
  • \( b \): Growth factor of selling scoops (2).

• For Graph 2:

  • \( x \): Number of ideal sunny days from start.
  • \( y \): Number of scoops sold.
  • \( a \): Initial scoops sold per day (18).
  • \( b \): Growth factor representing sales on sunny days (3).

E. Evaluating The Equations

For Graph 1:

Let’s evaluate \( y = 3 \cdot 2^{x} \) at \( x = 3 \):
\( y = 3 \cdot 2^3 = 3 \cdot 8 = 24 \).

The point (3, 24) means that on the third warmer day, the shop sold 24 scoops.

For Graph 2:

Let’s evaluate \( y = 18 \cdot 3^x \) at \( x = 3 \):
\( y = 18 \cdot 3^3 = 18 \cdot 27 = 486 \).

The point (3, 486) reflects how many scoops could be sold on the third sunny day.

F. Unique Exponential Growth Situation

Situation: The “Sundae Sundays” special at the ice cream shop starts with a sale of 12 sundaes and triples every following Sunday.

1. Exponential Model:

   • \( y = 12 \cdot 3^x \)

2. Input-Output Table:

| x | y | |----|-----| | 0 | 12 | | 1 | 36 | | 2 | 108 | | 3 | 324 |

3. Graph:

You would plot these points on a graph with \( x \) as the number of Sundays and \( y \) as the number of sundaes.

This example captures the excitement and growth of sales during a promotional event, highlighting how customer interest can echo through exponential growth.

In all situations, the base reflects customer engagement growth while the initial value is the starting sales figure.