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)

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

Part A: 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). To understand why this is the case, let's break it down step by step.

The equation y = 2^x represents exponential growth with base 2. As x increases, the value of 2^x increases exponentially.

The equation y = 4^(x-2) represents exponential growth with base 4. Since the exponent is (x-2), the function is shifted to the right by 2 units compared to the first equation. As x increases, the value of 4^(x-2) also increases exponentially.

When these two functions intersect, it means that at those specific points, the y-values from both equations are equal. Mathematically, we can express this as 2^x = 4^(x-2).

Part B: To find the solutions to the equation 2^x = 4^(x-2), we can create tables to explore different integer values of x between -4 and 4. Here's how we can do it:

For each integer value of x between -4 and 4, substitute the value into the equation and calculate the resulting y-values.

For example, let's start with x = -4:
2^(-4) = 4^(-4-2) = 1/16
This gives us the point (-4, 1/16).

Repeat this process for the remaining integer values of x:

x = -3: 2^(-3) = 4^(-3-2) = 1/8
x = -2: 2^(-2) = 4^(-2-2) = 1/4
x = -1: 2^(-1) = 4^(-1-2) = 1/2
x = 0: 2^0 = 4^(0-2) = 1/4
x = 1: 2^1 = 4^(1-2) = 1/2
x = 2: 2^2 = 4^(2-2) = 1
x = 3: 2^3 = 4^(3-2) = 2
x = 4: 2^4 = 4^(4-2) = 4

The resulting table would look like this:

x | y
-4 | 1/16
-3 | 1/8
-2 | 1/4
-1 | 1/2
0 | 1/4
1 | 1/2
2 | 1
3 | 2
4 | 4

Part C: To solve the equation 2^x = 4^(x-2) graphically, you can plot the graphs of both functions on the same coordinate system and look for points where they intersect.

First, plot the graph of y = 2^x, which represents exponential growth with a base of 2.

Then, plot the graph of y = 4^(x-2), which represents exponential growth with a base of 4. Remember to shift the second equation to the right by 2 units.

Once the graphs are plotted, look for points where they intersect. These points represent the x-coordinates of the solutions to the equation 2^x = 4^(x-2).