Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 4^–x and y = 2^x + 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

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

Answer: Given two simultaneous equations that are both required to be true.. the solution is the points where the lines cross... Which is where the two equations are equal.. Thus the solution that works for both equations is when 2-x = 4x+3 because where that is true is where the two lines will cross and that is the common point that satisfies both equations.
       
Question: Make tables to find the solution to 4^–x = 2^x + 3. Take the integer values of x between â’3 and 3. (4 points)
       
Answer: x 2-x 4x+3 ______________ -3 5 -9 -2 4 -5 -1 3 -1 0 2 3 1 1 7 2 0 11 3 -1 15 The table shows that none of the integers from [-3,3] work because in no case does 2-x = 4x+3 To find the solution we need to rearrange the equation to the form x=n 2-x = 4x+3 2 -x + x = 4x + x +3 2 = 5x + 3 2-3 = 5x +3-3 5x = -1 x = -1/5 The only point that satisfies both equations is where x = -1/5 Find y: y = 2-x = 2 - (-1/5) = 2 + 1/5 = 10/5 + 1/5 = 11/5 Verify we get the same in the other equation y = 4x + 3 = 4(-1/5) + 3 = -4/5 + 15/5 = 11/5 Thus the only actual solution, being the point where the lines cross, is the point (-1/5, 11/5)
       
Question: How can you solve the equation 4^–x = 2^x + 3 graphically? (2 points)
   
Answer: To solve graphically 2-x=4x+3 we would graph both lines... y = 2-x and y = 4x+3 The point on the graph where the lines cross is the solution to the system of equations ... [It should be, as shown above, the point (-1/5, 11/5)] To graph y = 2-x make a table.... We have already done this in part B x 2-x x 4x+3 _______ ________ -1 3 -1 -1 0 2 0 3 1 1 1 7 Just graph the points on a cartesian coordinate system and draw the two lines. The solution is, as stated, the point where the two lines cross on the graph.