Question

The graph of f(x) = 2x ^ 2 is given below. Write the equation of a function g(x) which would shift the graph of f(x) left 3 units and up 2 units.

Answer 1

To shift the graph of \( f(x) = 2x^2 \) left 3 units, you replace \( x \) with \( x + 3 \). To shift the graph up 2 units, you add 2 to the entire function.

Starting with \( f(x) \):

1. Shift left 3 units: \( f(x + 3) = 2(x + 3)^2 \)
2. Shift up 2 units: \( g(x) = 2(x + 3)^2 + 2 \)

Now, we can simplify \( g(x) \):

\[
g(x) = 2(x + 3)^2 + 2 = 2(x^2 + 6x + 9) + 2 = 2x^2 + 12x + 18 + 2 = 2x^2 + 12x + 20
\]

Therefore, the equation of the function \( g(x) \) is:

\[
g(x) = 2(x + 3)^2 + 2
\]

In its expanded form, it can also be written as:

\[
g(x) = 2x^2 + 12x + 20
\]