Question

What is the possible for $g\left( x \right)$ in terms of $f\left( x \right)$?
The graph of $g\left( x \right)$ is the graph of $f\left( x \right)$ after it has been vertically stretched or shrunk. The point $\left( 3,6 \right)$ lies on the graph of $f\left( x \right)$ . The corresponding point on the graph of $g\left( x \right)$ is $\left( 3,12 \right)$. What is the possible formula for $g\left( x \right)$ in terms of $f\left( x \right)$?

Answer 1

We can observe that the point shifts up by 6 units. If $k>1$, the graph of $y=k\centerdot f\left( x \right)$ is the graph of $f\left( x \right)$ vertically stretched by multiplying each of its $y-$coordinates by $k$. If $0 < k < 1$, the graph is $f\left( x \right)$ vertically shrunk by multiplying each of its $y-$ coordinates by $k$.

Complete step-by-step answer:
Now let us find out the possible formula for $g\left( x \right)$ in terms of $f\left( x \right)$.
Since $f\left( x \right)$ is vertically stretched or shrunk in a graph, we have to multiply a number $n$, which is greater than 1 if the graph is stretched else with a number less than 1 but greater than 0. This makes the graph shrink.
Contrarily, if we just add $a$, it will either shift it up if it is positive or shift it down, if it is negative.
Since our given condition is either stretched or shrunk, we get
$g\left( x \right)=2f\left( x \right)$
On verifying, we find that
If $f\left( x \right)=6$
$g\left( x \right)=2f\left( x \right)$
Then, $2\times 6=12$.
$\therefore$ The possible formula of $g\left( x \right)$ in terms of $f\left( x \right)$ is $g\left( x \right)=2f\left( x \right)$.

Note: We can notice that when the graph is stretched, we can expect that the graph’s $y-$coordinates to be farther from the $x-$ axis but the input values remains same. In general, the vertical stretch equation is given by $y=k\centerdot f\left( x \right)$.
The graph can be plotted in the following way-