Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 4–x and y = 2x + 3 intersect are the solutions of the equation 4^–x = 2^x + 3. (4 points)

Part B: Make tables to find the solution to 4^–x = 2^x + 3. Take the integer values of x between −3 and 3. (4 points)

Part C: How can you solve the equation 4^–x = 2^x + 3 graphically? (2 points)

Answer 1

A.
so
y=4⁻ˣ and y=2ˣ+3 can be combined to get
4⁻ˣ=y=2ˣ+3
4⁻ˣ=2ˣ+3
the x value that makes them both true is the same as the first  2 equations because they are the same equation

B.
for y=4⁻ˣ
(3,1/64)
(2,1/16)
(1,1/4)
(0,1)
(-1,4)
(-2,16)
(-3,64)

for y=2ˣ+3
(3,11)
(2,7)
(1,5)
(0,4)
(-1,3.5)
(-2,3.25)
(-3,3.125)

so at about x=-1


C. graph y=4⁻ˣ and y=2ˣ+3 and see the intersection

Answer 2

(A) The first graph is the set of all points where y = 8x. The second graph is the set of points where y = 2x + 2. If a point is on both graphs, then both equations will be true at the same time. So we will have 8x = y and y = 2x + 2, so 8x = 2x + 2.

(B)
x . 8x . 2x + 2
-3 . -24 . -4
-2 . -16 . -2
-1 . -8 . . 0
0 . . 0 . . 2
1 . . 8 . . 4
2 . 16 . . 6
3 . 24 . . 8

From this you can see that the solution is between 0 and 1. Using algebra we can solve it easily:
8x = 2x + 2
=> 6x = 2
=> x = 1/3.

(C) Graph the two lines from part A and find the x-coordinate of the intersection.