Question

which graph shows the solution to the equation \\(4^{x-3} = 8\\)?

Explanation:

Solve the exponential equation algebraically

Using the Solving Exponential Equations Algebraically knowledge point

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Set up the system of equations for graphical solution

Using the Graphical Solution of Systems knowledge point

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Analyze the graphs to find the correct system representation

We need to identify the graph that plots the system:

1. An exponential growth curve \(y = 4^{x-3}\) with a horizontal asymptote at \(y = 0\).
2. A horizontal line \(y = 8\).
3. An intersection point at \(x = 4.5, y = 8\).

Looking at the options:

• First graph: Shows the exponential curve \(y = 4^{x-3}\) and a horizontal line at \(y = -8\).
• Second graph: Shows a decreasing curve and a vertical line at \(x = -6\).
• Third graph: Shows the exponential curve \(y = 4^{x-3}\) and a horizontal line at \(y = 8\). The intersection occurs at \(x = 4.5\).
• Fourth graph: Shows the exponential curve and a nearly vertical line.

Therefore, the third graph correctly represents the system.

Answer:

• (A) Graph showing the exponential curve and a horizontal line at \(y = -8\)
• (B) Graph showing a decreasing curve and a vertical line at \(x = -6\)
• (C) Graph showing the exponential curve and a horizontal line at \(y = 8\) (Correct answer)
• (D) Graph showing the exponential curve and a nearly vertical line