Value of `y` increases exponentially from `y =- 4` to `y = + 4`. Plot `a x-y` graph.

Answer

To solve the problem of plotting an exponential graph where the value of \( y \) increases from \( y = -4 \) to \( y = +4 \), we can follow these steps:

### Step 1: Understand the Exponential Function
The exponential function can be represented as:
\[ y = e^{ax} \]
where \( a \) is a constant that determines the rate of growth. In this case, we need to find the values of \( x \) that correspond to \( y = -4 \) and \( y = +4 \).

### Step 2: Set Up the Equations
Given that \( y \) increases from \( -4 \) to \( +4 \), we can express this in terms of the exponential function:
1. At \( x = x_1 \), \( y = -4 \):
\[ -4 = e^{ax_1} \]
However, since the exponential function \( e^{ax} \) is always positive, we cannot have \( y = -4 \) in this context. Therefore, we need to adjust our approach.

2. Instead, we can consider a transformation of the exponential function to fit the range from negative to positive values. A common approach is to use:
\[ y = A(e^{ax} - 1) \]
where \( A \) is a scaling factor.

### Step 3: Determine the Scaling Factor
To find the scaling factor \( A \), we need to set the limits:
- When \( x = x_1 \), \( y = -4 \):
\[ -4 = A(e^{ax_1} - 1) \]
- When \( x = x_2 \), \( y = +4 \):
\[ 4 = A(e^{ax_2} - 1) \]

### Step 4: Solve for \( A \)
From the two equations, we can express \( A \) in terms of \( e^{ax_1} \) and \( e^{ax_2} \):
1. From the first equation:
\[ A = \frac{-4}{e^{ax_1} - 1} \]
2. From the second equation:
\[ A = \frac{4}{e^{ax_2} - 1} \]

### Step 5: Plot the Graph
Now that we have the equations set up, we can plot the graph:
1. Choose values for \( x_1 \) and \( x_2 \) such that \( e^{ax_1} \) and \( e^{ax_2} \) yield reasonable values.
2. Plot the points corresponding to \( y = -4 \) and \( y = +4 \).
3. Draw the curve that connects these points, which will show an exponential increase.

### Step 6: Finalize the Graph
The graph will show an increasing curve starting from \( y = -4 \) and approaching \( y = +4 \) as \( x \) increases.

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