Question

PLEASE ANSWER QUICK ITS A TIMED QUIZ Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 4^x and y = 2^x−2 intersect are the solutions of the equation 4^x = 2^x−2. (4 points) Part B: Make tables to find the solution to 4^x = 2^x−2. Take the integer values of x between −3 and 3. (4 points) Part C: How can you solve the equation 4^x = 2^x−2 graphically? (2 poin

Answer 1

A.  We have two lines:  y = 4-x   and   y = 8-x^-1

Given two simultaneous equations that are both to be true, then the solution is the points where the lines cross. The intersection is where the two equations are equal. Therefore the solution that works for both equations is when

4-x = 8-x^-1

This is where the two lines will cross and that is the common point that satisfies both equations.

 

B.  4-x = 8-x^-1

 

 x         4-x      8-x^-1

______________

 

-3          7        8.33

-2          6        8.5

-1          5        9

 0          4        -

 1          3        7

 2          2        7.5

 3          1        7.67

 

The table shows that none of the x values from -3 to 3 is the solution because in no case does

4-x = 8-x^-1

 

To find the solution we need to rearrange the equation to find for x:

4-x = 8-x^-1

Multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

 

Therefore there are two points that satisfies the equation.

Find y:  

x=-4.236

y = 4-x  = 4 – (-4.236) = 8.236

y = 8-x^-1 =  8-(-4.236)^-1 = 8.236

 

x=0.236

 y = 4-x  = 4 – (0.236) = 3.764

y = 8-x^-1 =  8-(0.236)^-1 = 3.764

 

Thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

 

C.  To solve graphically the equation 4-x = 8-x^-1

We would graph both lines: y = 4-x  and   y = 8-x^-1

The point on the graph where the lines cross is the solution to the system of equations.

Just graph the points on part B on a Cartesian coordinate system and extend the two lines.  The solution is, as stated, the point where the two lines cross on the graph.