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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.

3 answers

To answer your questions regarding the exponential graphs indicated by the points you provided, let's consider a general format for a typical exponential function equation:

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

Where:

  • \( y \) is the output,
  • \( x \) is the input,
  • \( a \) is the initial value (value of \( y \) when \( x = 0 \)),
  • \( b \) is the base, which indicates the rate of growth.

A. Input-Output Tables

Table for Graph 1:

  • Let's assume the points are: (-2, 3/4), (1, 6)
  • Assume additional points, say (0, 1) and (2, 12)

| x | y | |-----|--------| | -2 | 3/4 | | 0 | 1 | | 1 | 6 | | 2 | 12 |

Table for Graph 2:

  • Points: (-2, 2/9), (1, 6)
  • Assume additional points, say (0, 1) and (2, 12)

| x | y | |-----|--------| | -2 | 2/9 | | 0 | 1 | | 1 | 6 | | 2 | 12 |

B. Exponential Equations

Graph 1:

  1. Initial value (when \( x = 0 \)): \( a = 1 \)
  2. Base \( b \) can be found using two points (0, 1) and (1, 6): \[ 6 = 1 \cdot b^1 \implies b = 6 \] Hence, the equation is: \[ y = 1 \cdot 6^x \]

Graph 2:

  1. Initial value: \( a = 1 \)
  2. Using the points (0, 1) and (1, 6): \[ 6 = 1 \cdot b^1 \implies b = 6 \] Hence, the equation is: \[ y = 1 \cdot 6^x \]

C. Real-World Situations

Graph 1:

  • A population of bacteria that grows exponentially. Initially, there is one bacterium, and it doubles in size every hour.

Graph 2:

  • A financial investment that grows at a rate of 500% in the first year after an initial deposit of $1.

D. Explanation of Variables

For both situations:

  • \( x \): Time in hours (for the bacteria) or years (for the financial investment).
  • \( y \): Population of bacteria or the value of the investment.
  • \( a \): The initial amount (1 bacterium or $1).
  • \( b \): The growth factor (in this case, it seems to be the same for both graphs, being 6).

E. Evaluate for New Values of x

Let's evaluate for \( x = 3 \):

  1. For the exponential equation \( y = 1 \cdot 6^x \): \[ y = 6^3 = 216 \]
    • This means after 3 hours, there would be 216 bacteria, or after 3 years the investment would be worth $216.

F. Unique Growth Situation

Situation: A tree grows such that its height increases exponentially at a rate of 150% per year, starting at a height of 1 meter.

Equation: \[ y = 1 \cdot 1.5^x \]

Input-Output Table:

| x (Years) | y (Height in Meters) | |-----------|-----------------------| | 0 | 1 | | 1 | 1.5 | | 2 | 2.25 | | 3 | 3.375 |

Graph: This graph would show a curve that starts at 1 meter and rises steeply as x increases.

This encompasses all the requested features for your task! Please make sure to adjust any values or details based on the specific features or points of your graphs if they differ.

Can you redo it all but the real world situation is about an ice cream shop

Sure! Let’s rework everything to focus on an ice cream shop scenario.

A. Input-Output Tables

Table for Graph 1:

  • Given points: (-2, 3/4), (1, 6)
  • Additional assumed points: (0, 1.5), (2, 12)

| x | y | |-----|--------| | -2 | 3/4 | | 0 | 1.5 | | 1 | 6 | | 2 | 12 |

Table for Graph 2:

  • Given points: (-2, 2/9), (1, 6)
  • Additional assumed points: (0, 1.5), (2, 12)

| x | y | |-----|---------| | -2 | 2/9 | | 0 | 1.5 | | 1 | 6 | | 2 | 12 |

B. Exponential Equations

Graph 1:

  • Initial value \( a = 1.5 \) (at \( x = 0 \)).
  • To find the base \( b \), using the two points (0, 1.5) and (1, 6): \[ 6 = 1.5 \cdot b^1 \implies b = \frac{6}{1.5} = 4 \] Hence, the equation is: \[ y = 1.5 \cdot 4^x \]

Graph 2:

  • Initial value \( a = 1.5 \) (at \( x = 0 \)).
  • Using the points (0, 1.5) and (1, 6): \[ 6 = 1.5 \cdot b^1 \implies b = \frac{6}{1.5} = 4 \] Hence, the equation is: \[ y = 1.5 \cdot 4^x \]

C. Real-World Situations

Graph 1:

  • An ice cream shop starts with an initial sales of 1.5 liters of ice cream sold (maybe during a slow day), and every hour their sales quadruples due to a new marketing strategy or an influx of customers.

Graph 2:

  • The same ice cream shop begins with 1.5 liters in sales on a particular slow day. After one hour, their sales increase to 6 liters; the shop has a promotion that causes sales to quadruple each hour.

D. Explanation of Variables

In both cases:

  • \( x \): Time in hours since the promotional event started.
  • \( y \): Liters of ice cream sold.
  • \( a \): Initial sales amount (1.5 liters).
  • \( b \): The growth factor, which is 4 (indicating that sales quadruple every hour).

E. Evaluate for New Values of x

Let's evaluate for \( x = 3 \):

Using the equation \( y = 1.5 \cdot 4^x \): \[ y = 1.5 \cdot 4^3 = 1.5 \cdot 64 = 96 \]

  • This means after 3 hours, the ice cream shop could potentially sell 96 liters of ice cream if the trend continues.

F. Unique Ice Cream Shop Situation

Situation: An ice cream shop has a special event that starts with an initial sale of 1.5 liters of a unique flavor and doubles its sales each hour due to high customer interest.

Equation: \[ y = 1.5 \cdot 2^x \]

Input-Output Table:

| x (Hours) | y (Liters of Ice Cream Sold) | |-----------|-------------------------------| | 0 | 1.5 | | 1 | 3 | | 2 | 6 | | 3 | 12 |

Graph: This graph would start at 1.5 liters and rise steeply, reflecting the doubling sales.

This reworked response captures the ice cream shop theme, presenting clear exponential behavior in sales growth. Feel free to adjust any details as necessary!