To fill in the missing elements for the exponential growth equation \( y = a \cdot b^x \), we need to determine the values of \( a \) and \( b \) based on the graph you've described. Since I cannot see the specific marked points on the graph, I will give you a general way to find \( a \) and \( b \):
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Identify a point on the curve (x, y): Look for the coordinates of the point where the curve intersects the graph. This can be your first point.
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Identify another point on the curve as a second data point (x', y'): Find at least one more point on the curve to help determine the growth rate.
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Determine \( a \): The value of \( a \) represents the initial value when \( x = 0 \). If there is a point where \( x = 0 \) on your graph, \( a \) is simply the \( y \)-coordinate of that point.
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Calculate \( b \): To calculate \( b \) using the points (x, y) and (x', y'), you can rearrange the equation: \[ \frac{y'}{y} = b^{x' - x} \] From this equation, you can solve for \( b \): \[ b = \left( \frac{y'}{y} \right)^{\frac{1}{x' - x}} \]
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Complete the equation: Once you have \( a \) and \( b \), substitute these into the formula \( y = a \cdot b^x \).
If you can provide the specific \( x \) and \( y \) values from the points marked on the curve, I can help you determine the specific values for \( a \) and \( b \).