Question

How do you graph the equation \[y=5{{\left( 2 \right)}^{x}}\].

Answer 1

Hint: The given problem is an exponential function. An exponential function is a function in which a number or a constant ratio is raised to a variable. An exponential that goes up from left to right is called exponential growth. We know that the equation of an exponential function is \[y=a.{{b}^{x}}+c\]. In this problem, an exponential equation is given and we have to draw a graph for that given function.

Complete step by step answer:
We know that the given exponential function is,
 \[y=5{{\left( 2 \right)}^{x}}\]…….. (1)
We also know that the general equation of an exponential function is
\[y=a.{{b}^{x}}+c\]……. (2)
Where a is the y-intercept, b is the constant ratio, c is the horizontal asymptote.
Now we can compare equation (1) and equation (2),
We get,
y-intercept at y, a =5
constant ratio, b=2
horizontal asymptote at y, c=0.
Now we have to find the significant points,
When x=0, from equation (1) we have,
\[\begin{align}
  & y={{5.2}^{0}} \\
 & \Rightarrow y=5\text{ }\because {{2}^{0}}=1 \\
\end{align}\]
So, y axis intercept is at \[\left( 0,5 \right)\].
if x<0 we have,
\[\begin{align}
  & \Rightarrow \dfrac{5}{{{2}^{x}}},x\to \infty \\
 & \Rightarrow \dfrac{5}{{{2}^{\infty }}}\to 0 \\
\end{align}\]
So, the x axis is a horizontal asymptote.
If x>0 we have,
\[\begin{align}
  & \Rightarrow 5{{\left( 2 \right)}^{x}},x\to \infty \\
 & \Rightarrow 5{{\left( 2 \right)}^{\infty }}\to \infty \\
\end{align}\]
Now we can draw the graph,

Note:
Students may find difficulties in understanding the concept of exponential functions graph. In this problem, we just have to find the y-intercept to plot the graph. In an exponential function graph, where the graph increases, and it continues. Each graph in the exponential function has a horizontal asymptote, which can be identified in the given function.