To determine which equation is likely modeled by the graph on the coordinate plane, we need to analyze the behavior and characteristics of the curve.
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End Behavior: The curve has arrows at both ends, which suggests that it is an exponential function. If it increases toward positive infinity as \(x\) goes to positive infinity, it indicates a positive exponential growth.
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Y-Intercept: The value of \(y\) when \(x = 0\) (the y-intercept) is often crucial. For functions in the form \(y = a(4)^x\), \(a\) will determine the initial value when \(x=0\), which will be \(y = a\).
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Exponential Growth Factors: The exponent base \(4\) indicates a rapid growth of the function, especially in positive \(x\) values.
Now, let’s examine the given options considering typical exponential function behavior:
- Option 1: \(y=3.5(4)^x\)
- Option 2: \(y=50(4)^x\)
- Option 3: \(y=-3.5(4)^x\) (not possible, as it would yield negative values)
- Option 4: \(y=10(4)^x\)
Since \(y\) appears to increase, we can discard Option 3. Among the positive options, we need to look at the substantial growth indicated by large initial values.
If \(y=50(4)^x\) truly corresponds to a steep and significant increase from a higher initial value, it’s likely the best fit since it has the largest initial multiplier. Both Options 1 and 4 have smaller starting values.
Based on these considerations, the most reasonable conclusion is that the equation modeled by the graph is:
y=50(4)^x.