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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.

1 answer

Based on the provided points, let’s create input-output tables, exponential equations, real-world situations, and additional explanations. I'm making an assumption about what the graphs may playout based on the provided coordinates.

A. Input-Output Tables

Assuming we have two graphs based on the points given, let’s make a rough layout:

Graph 1 | Input (x) | Output (y) | |-----------|------------| | -2 | 3/4 | | 0 | 1 | | 1 | 6 | | 2 | 36 |

Graph 2 | Input (x) | Output (y) | |-----------|------------| | -2 | 2/9 | | 0 | 1 | | 1 | 6 | | 2 | 36 |

B. Exponential Equations

For each graph, the general form of an exponential function is:

\[ y = a \cdot b^x \]

Graph 1:

  • From (0, 1), identify that initial value \( a = 1 \) (this is the \( y \)-intercept).
  • From (1, 6), we find the base \( b \): \[ 1 \cdot b^1 = 6 \Rightarrow b = 6 \]
  • Equation: \( y = 1 \cdot 6^x \) or simply \( y = 6^x \)

Graph 2:

  • From (0, 1), again the initial value \( a = 1 \).
  • From (1, 6): \[ 1 \cdot b^1 = 6 \Rightarrow b = 6 \]
  • Equation: \( y = 1 \cdot 6^x \) or \( y = 6^x \)

C. Real-World Situations

  1. Graph 1: Suppose you are studying a species of bacteria that doubles in number every hour. The bacteria population starts with 1 bacterium at hour zero. After one hour, the population reaches 6.

  2. Graph 2: Imagine you're investing in a savings account that grows by a factor of 6 every year. You start with $1 and after one year, it grows to $6.

D. Explanation of Variables

  • x: Represents time (days, hours, or years passed).
  • y: Represents the population of bacteria or the amount of money in the savings account.
  • a (initial value): Starting quantity (number of bacteria or initial investment).
  • b (growth factor): The multiplier indicating how much the population/investment grows each time unit.

E. Evaluation at a Value not Listed in the Table

Let's evaluate the equation for \( x = -1 \) for both graphs:

  1. For Graph 1: \[ y = 6^{-1} = \frac{1}{6} \approx 0.1667 \]

    • Meaning at one hour before starting the experiment, you would expect about 0.1667 bacteria, indicating less than one bacterium is present.

  2. For Graph 2: \[ y = 6^{-1} = \frac{1}{6} \approx 0.1667 \]

    • Interpreted similarly, this means one year prior, your investment would hypothetically yield about $0.1667.

F. Unique Exponential Growth Situation

Situation: A water tank that is filled to a certain capacity. The water level in the tank doubles every month due to an automatic filling mechanism. The tank initially starts with 2 liters of water.

Model it with an Equation

  1. Equation: Letting \( a = 2 \) liters and \( b = 2 \) (the water doubles each month), we have: \[ y = 2 \cdot 2^x \]

  2. Input-Output Table

| Month (x) | Water Level (y) | |-----------|------------------| | 0 | 2 | | 1 | 4 | | 2 | 8 | | 3 | 16 |

Graph

To graph this, plot points from the table and connect them to reflect the rapid growth of water level.

This comprehensive view provides equations, tables, and situational examples for understanding exponential growth in real-world contexts. If you have a specific variable or parameter you want to adjust, feel free to ask!