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

Answer:

A and B. See explanation

C. x=4

Step-by-step explanation:

A: Consider two functions $y=2^x$ and $y=4^{x-2}.$ The point of intersection of the graphs of these two functions is the solution of the system $\left \{ {{y=2^x} \atop {y=4^{x-2}}} \right.$

When you are solving this system, you have to equate the right sides of these equations and get $2^x=4^{x-2}.$ So, the x-coordinate of the point of intersection is the solution of the equation $2^x=4^{x-2}.$

B.

$\begin{array}{ccccccc}x&-1&0&1&2&3&4\\2^x&\frac{1}{2}&1&2&4&8&16\end{array}$

$\begin{array}{ccccccc}x&-1&0&1&2&3&4\\4^{x-2}&\frac{1}{64}&-\dfrac{1}{16}&\dfrac{1}{4}&1&4&16\end{array}$

C. Graphically, graphs intersect at point (4,16), so x=4 is the solution of the equation