Question

standard 10 homework
score: 13/24 answered: 11/20
question 12
draw the graph of $f(x)=5^{x + 2}$
clear all draw:
question help: video message instructor post to forum

Explanation:

Step1: Find some points

Let's choose some values of \(x\) to calculate \(y = f(x)=5^{x + 2}\).
When \(x=-2\), \(y = 5^{-2+2}=5^{0}=1\).
When \(x=-1\), \(y = 5^{-1 + 2}=5^{1}=5\).
When \(x = 0\), \(y=5^{0+2}=25\).

Step2: Analyze the function properties

The general form of an exponential function is \(y = a^{x + h}\), where \(a = 5>1\). The function \(y = 5^{x+2}\) is an exponential growth function. The graph of \(y = 5^{x}\) is shifted \(2\) units to the left to get the graph of \(y = 5^{x + 2}\).

Step3: Plot the points and draw the graph

Plot the points \((-2,1)\), \((-1,5)\), \((0,25)\) on the coordinate plane. Then, since it is an exponential growth function (\(a>1\)), as \(x\to-\infty\), \(y\to0\) (the \(x\) - axis is a horizontal asymptote \(y = 0\)), and as \(x\to+\infty\), \(y\to+\infty\). Connect the points with a smooth curve.

Answer:

Plot the points \((-2,1)\), \((-1,5)\), \((0,25)\) and draw a smooth exponential growth curve approaching the \(x\) - axis as \(x\to-\infty\) and increasing without bound as \(x\to+\infty\).