Question

Part A: explain why x coordinates of the points where the graphs of the equations y=4^x and y=2^(x-1) intersect are the solutions of the equation 4^x=2^(x-1). Part B: Make tables to find the solution to 4^x=2^(x-1). Take the integer values of x between -4 and 4. Part C: How can you solve the equation 4^x=2^(x-1) graphically?

Answer 1

9514 1404 393

Explanation:

A. The given equation will have a solution when the value of x makes the left-side expression equal to the right-side expression.

The value of the left-side expression for a given value of x is the y-value of y=4^x. The value of the right-side expression for a given value of x is the y-value of y=2^(x-1). The left-side expression will be equal to the right-side expression for a given value of x when the graph of y=4^x and the graph of y=2^(x-1) have the same y-value. That is, the graphs will be of the same point for that x-value. That is what we mean when we say the curves will intersect at that point.

The x-coordinate of the point of intersection is the value that makes the expressions equal, hence it is the solution to the equation.

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B. The tables are included in the first attachment. We have used f(x) to represent the left-side expression, and g(x) to represent the right-side expression.

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C. As described in Part A, the graphs will cross at the solution point. Hence the equation can be solved graphically by graphing the left-side expression and the right-side expression and seeing where the graphs cross. (That is at the point (-1, 0.25) on the attached graph.)

Alternate graphical method

The solution can also be found graphically by graphing the difference of the expressions: y = 4^x -2^(x-1). This difference will be zero at the value of x that makes the two expressions equal. That is, the x-intercept of the graph of this difference will be the solution value. The second attachment shows this sort of graphical solution. (It works nicely because many graphing calculators will display the value of the x-intercept.)