Ask a New Question
Menu

Use the graphs to answer the questions.

(-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

1 answer

To address your request, let's break it down step by step for better understanding and clarity regarding exponential growth, focusing on an ice cream shop scenario.

A. Input-Output Tables

  1. Graph 1 (Based on points provided)

    • Input-Output Table:
      |  x  |  y   |
  |-----|------|
  | -2  | 0.75 |
  | -1  | 1.5  |
  |  0  | 3    |
  |  1  | 6    |

  2. Graph 2 (Points indicated)

    • Input-Output Table:
      |  x  |  y   |
  |-----|------|
  | -2  | 0.25 |
  | -1  | 0.9  |
  |  0  | 1    |
  |  1  | 6    |

B. Exponential Equations

  1. Graph 1: Let's say the general form of our exponential function is \( y = a \cdot b^x \).

    • From the table, let's assume:
      • At \( x = 0 \), \( y = 3 \) (initial value).
      • The growth seems to increase as \( x \) increases; hence, we can assume \( b = 2 \).
    • So the equation is: \( y = 3 \cdot 2^x \).

  2. Graph 2: Similarly, we can define:

    • Initial value at \( x = 0 \) is 1.
    • The growth is slower, so let’s say \( b = 3 \).
    • Therefore, the equation could be: \( y = 1 \cdot 3^x \).

C. Real-World Situations

  1. Graph 1: An ice cream shop starts with 3 different flavors available for customers. Each month, the shop doubles the number of flavors offered.

  2. Graph 2: An ice cream shop starts with a basic menu of 1 unique flavor, and every month they manage to introduce 3 new special flavors.

D. Explanation of Variables

  • In both equations:
    • \( x \) represents the time in months.
    • \( y \) represents the number of flavors available.
    • \( a \) is the initial number of flavors.
    • \( b \) is the factor by which the number of flavors grows each month.

E. Evaluating the Equations

  1. For Graph 1:

    • If we evaluate \( x = 3 \):
      • \( y = 3 \cdot 2^3 = 3 \cdot 8 = 24 \).
      • This means that after 3 months, the shop could offer 24 different flavors.

  2. For Graph 2:

    • If we evaluate \( x = 3 \):
      • \( y = 1 \cdot 3^3 = 27 \).
      • So after 3 months, the shop would have 27 unique flavors available.

F. Unique Exponential Growth Situation: Ice Cream Shop

Situation: An ice cream shop, "Sweet Treats," starts with 5 flavors and finds that every week, they can successfully add 1 more unique flavor from customer suggestions.

Modeling with an Equation:

  • The initial value (at \( x = 0 \)) is \( 5 \).
  • The growth is represented as an additive process every week but can be modeled exponentially for simplification.
  • Equation: \( y = 5 \cdot 1.1^x \)
    • Where \( b \) is slightly adjusted for the context of new flavors gaining popularity (a 10% increase each week).

Input-Output Table:

     |  x  |  y   |
     |-----|------|
     |  0  |  5   |
     |  1  |  5.5 |
     |  2  |  6.05|
     |  3  |  6.655|

Graph: The exponential growth curve will rise gradually as the weeks pass due to the consistent addition of flavors.

This completes the model and explanation of each part of your request centered on an ice cream shop, incorporating mathematical concepts related to exponential growth!