Question

which of the following graphs could be used to determine the solution to the equation 2 = 2·6ˣ?

Explanation:

Step1: Analyze the equation

To solve \(2 = 2\cdot6^{x}\) graphically, we can rewrite it as finding the intersection of \(y = 2\) (a horizontal line) and \(y=2\cdot6^{x}\) (an exponential function).

Step2: Analyze the exponential function \(y = 2\cdot6^{x}\)

• When \(x = 0\), \(y=2\cdot6^{0}=2\cdot1 = 2\). So the exponential function \(y = 2\cdot6^{x}\) passes through the point \((0, 2)\).
• The horizontal line \(y = 2\) is a horizontal line parallel to the \(x -\)axis passing through \((x, 2)\) for all \(x\).

Step3: Analyze the graphs

• For the first graph: The exponential curve passes through \((0, 2)\) (since when \(x = 0\), \(y = 2\)) and there is a horizontal line \(y = 2\). The intersection of \(y = 2\) and \(y=2\cdot6^{x}\) will be at \(x = 0\) (because \(2=2\cdot6^{0}\)), and this graph has the horizontal line \(y = 2\) and the exponential curve starting at \((0, 2)\) (or near) and increasing.
• For the second graph: The exponential curve at \(x = 0\) has \(y=2\cdot6^{0}=2\)? Wait, no, looking at the second graph, when \(x = 0\), the \(y -\)value is around \(6\) (from the graph), so \(y=2\cdot6^{0}=2\) does not match, so this graph is for a different exponential function (maybe \(y = 6^{x}\) or \(y = 5\cdot6^{x}\) etc.).
• For the third graph: The horizontal line is at \(y = 4\) (not \(y = 2\)), so it can't be used.
• For the fourth graph: The horizontal line is at \(y = 1\) (or lower) and the exponential curve at \(x = 0\) is at \(y = 1\) (not \(2\)), so it's not matching.

Answer:

The first graph (the one with the horizontal line \(y = 2\) and the exponential curve passing through \((0, 2)\))