Question

Consider the function f(x) = p(0.5)" + where p and q are constants. The graph of f(x) passes through the points (0.6) and (1,4) and is shown below. a. Use the equations 6 = p+q to find the values of p and of q. 4 = 0.5p + 9 b. Write down the equation of the exponential function b. Write down the equation of the horizontal asymptote to the graph of f(x).

Answer 1

`\begin{gathered} a)\text{ p = 4, q = 2} \\ b)f(x)=4(0.5)^x\text{ + 2} \\ c)\text{ y = 2} \end{gathered}`

a) We are to use the given system of linear equations to find the value of p and q

To find these values, we have to solve the system of linear equations simultaneously

We have this as follows;

`\begin{gathered} 6\text{ = p + q} \\ 4\text{ = 0.5p + q} \\ \text{from second equation;} \\ q\text{ = 4-0.5p} \\ \text{Put this into first equation} \\ 6\text{ = p + 4-0.5p} \\ 6-4\text{ = p-0.5p} \\ 0.5p\text{ = 2} \\ p\text{ = }(2)/(0.5) \\ p\text{ = 4} \end{gathered}`

To get the value of q, we simply make a substitution for the value of q in any of the initial equations

We have this as;

`\begin{gathered} 6\text{ =p + q} \\ 6=\text{ 4+q} \\ q\text{ = 6-4} \\ q\text{ = 2} \end{gathered}`

b) Using the values of p and q obtained, we are to write down the equation for the exponential function

We have this as follows;

`f(x)=4(0.5)^x\text{ + 2}`

c) Here, we want to write the equation of the horizontal asymptote to the graph of f(x)

Mathematically, the asymptote is that line in which the graph approaches but will not touch as it moves away from the origin of the plot

With respect to the given question, we have this as follows;

`y\text{ = 2}`