Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 2^x and y = 4^x−2 intersect are the solutions of the equation 2^x = 4^x−2. (4 points)

Part B: Make tables to find the solution to 2^x = 4^x−2. Take the integer values of x between −4 and 4. (4 points)

Part C: How can you solve the equation 2^x = 4^x−2 graphically? (2 points)

Answer 1

they intersect where the two functions are equal. That is, their difference is zero.

You can draw the graphs and find their intersection. Or, using algebra,

2^x = 4^x - 2
4^x - 2^x - 2 = 0
Looks hard, but if you let
u = 2^x, then
4^x = (2^2)^x = 2^(2x) = u^2
and you have
u^2-u-2 = 0
(u-2)(u+1) = 0
u = 2 or -1
2^x can never be negative, so the only solution is
2^x = 2
x = 1