Asked by molo
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)?
Answers
Answered by
GPT-4o mini
To shift the function \( f(x) = 2x^2 - 1 \) to the left by 3 units, you need to replace \( x \) with \( x + 3 \) in the original function. This leads to the new function \( g(x) \).
Here's how it's done:
\[
g(x) = f(x + 3) = 2(x + 3)^2 - 1
\]
Next, we can expand this expression:
\[
g(x) = 2((x + 3)(x + 3)) - 1
\]
\[
= 2(x^2 + 6x + 9) - 1
\]
\[
= 2x^2 + 12x + 18 - 1
\]
\[
= 2x^2 + 12x + 17
\]
Thus, the expression representing \( g(x) \) is:
\[
g(x) = 2x^2 + 12x + 17
\]
Here's how it's done:
\[
g(x) = f(x + 3) = 2(x + 3)^2 - 1
\]
Next, we can expand this expression:
\[
g(x) = 2((x + 3)(x + 3)) - 1
\]
\[
= 2(x^2 + 6x + 9) - 1
\]
\[
= 2x^2 + 12x + 18 - 1
\]
\[
= 2x^2 + 12x + 17
\]
Thus, the expression representing \( g(x) \) is:
\[
g(x) = 2x^2 + 12x + 17
\]
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