if the original function f(x)=2x^2-1 is shifted to the left 3 units to make the function g(x), which expression would represent g(x)?

Question

2(x-3)^2-1
2(x+3)^2-1
2x^2+2
2x^2-4

Answer 1

To shift a function to the left by a certain number of units, you replace \(x\) with \(x + k\), where \(k\) is the number of units you want to shift (in this case, 3 units), so you replace \(x\) with \(x + 3\).

Starting with the original function:

\[ f(x) = 2x^2 - 1 \]

To shift it to the left by 3 units, we modify \(x\) as follows:

\[ g(x) = f(x + 3) = 2(x + 3)^2 - 1 \]

Now, we need to evaluate which option corresponds to \(g(x)\):

1. \( 2(x - 3)^2 - 1 \) (This shifts the function to the right, not the left)
2. \( 2(x + 3)^2 - 1 \) (This shifts the function to the left by 3 units)
3. \( 2x^2 + 2 \) (This changes the structure of the function)
4. \( 2x^2 - 4 \) (This changes the value but not the position)

The correct expression representing \(g(x)\) after shifting \(f(x)\) to the left by 3 units is:

\[ \boxed{2(x + 3)^2 - 1} \]