To solve an exponential equation using the given points and confirm the function, we start by understanding the structure of the equation. The general form of an exponential function can be written as:

$y = a \cdot e^{(bx + c)} + d$

Given points:

1. $(85.27772, 0)$
2. $(0, 27.5917)$
3. $(75.34961, 10)$
4. $(41.41311, 24.15377)$

We need to find the constants $a$, $b$, $c$, and $d$.

Step 1: Analyzing the provided function

The function provided for verification is: $y = -98.4 \cdot e^{(0.044x - 5)} + 28.254716$

This can be rewritten in the form: $y = -98.4 \cdot e^{(0.044x) - 5} + 28.254716$ Here, $a = -98.4$, $b = 0.044$, $c = -5$, and $d = 28.254716$.

Step 2: Verify the function using the given points

We will plug in each of the points into the function $y = -98.4 \cdot e^{(0.044x - 5)} + 28.254716$ and check if it holds true for each point.

1. For point $(85.27772, 0)$:

   $y = -98.4 \cdot e^{(0.044 \times 85.27772 - 5)} + 28.254716$ $y = -98.4 \cdot e^{(3.75181768 - 5)} + 28.254716$ $y = -98.4 \cdot e^{-1.24818232} + 28.254716$ $y \approx -98.4 \cdot 0.287 (e^{-1.24818232}) + 28.254716$ $y \approx 0$

2. For point $(0, 27.5917)$:

   $y = -98.4 \cdot e^{(0.044 \times 0 - 5)} + 28.254716$ $y = -98.4 \cdot e^{-5} + 28.254716$ $y \approx -98.4 \cdot 0.0067 (e^{-5}) + 28.254716$ $y \approx 27.5917$

3. For point $(75.34961, 10)$:

   $y = -98.4 \cdot e^{(0.044 \times 75.34961 - 5)} + 28.254716$ $y = -98.4 \cdot e^{(3.31498284 - 5)} + 28.254716$ $y = -98.4 \cdot e^{-1.68501716} + 28.254716$ $y \approx -98.4 \cdot 0.185 (e^{-1.68501716}) + 28.254716$ $y \approx 10$

4. For point $(41.41311, 24.15377)$:

   $y = -98.4 \cdot e^{(0.044 \times 41.41311 - 5)} + 28.254716$ $y = -98.4 \cdot e^{(1.82117684 - 5)} + 28.254716$ $y = -98.4 \cdot e^{-3.17882316} + 28.254716$ $y \approx -98.4 \cdot 0.041 (e^{-3.17882316}) + 28.254716$ $y \approx 24.15377$

Since the points satisfy the given function $y = -98.4 \cdot e^{(0.044x - 5)} + 28.254716$, this confirms that the equation is correct.

Do you want details or have any questions?

Here are 5 related questions:

1. How can we solve an exponential equation if only two points are given?
2. What is the significance of the constant $c$ in the exponential equation?
3. How does the value of $a$ affect the shape of the exponential curve?
4. Can exponential models be used to represent real-world phenomena? If so, what are some examples?
5. What happens to the graph of an exponential function when the value of $b$ changes?

Tip: Exponential equations often describe processes involving growth or decay, such as population dynamics, radioactive decay, or interest calculations.