Given:
The given parent function is [tex]f(x)=x^2[/tex]
We need to determine the equation of the new translated (shifted) function g(x).
Vertical stretch:
The general rule to shift the graph f(x), to shift c units upward is [tex]g(x)=f(x)+c[/tex]
From the graph, it is obvious that the graph f(x) is shifted 1 unit upwards.
Thus, applying the above rule, we get;
[tex]g(x)=x^2+1[/tex]
Horizontal stretch:
The general rule to shift the graph f(x) to shift c units to the left is [tex]g(x)=f(x+c)[/tex]
From, the graph, it is obvious that the graph f(x) is shifted 2 units to the left.
Thus, applying the above rule, we have;
[tex]g(x)=(x+2)^2[/tex]
Equation of the new function g(x):
From the figure, it is obvious that the graph g(x) is shifted 1 unit upwards and 2 units to the left.
Thus, we have;
[tex]g(x)=(x+2)^2+1[/tex]
Therefore, the equation of the new function g(x) is [tex]g(x)=(x+2)^2+1[/tex]
