To determine $g(x-1)$ for the function $g(x) = -5x^2 + 6x + 7$, substitute $x-1$ for $x$ in the original function.

Given: $g(x) = -5x^2 + 6x + 7$

1. Substitute $x-1$ in place of $x$: $g(x-1) = -5(x-1)^2 + 6(x-1) + 7$

2. Expand each term:

   • Start with $(x-1)^2$: $(x-1)^2 = x^2 - 2x + 1$
   • Substitute into $g(x-1)$: $g(x-1) = -5(x^2 - 2x + 1) + 6(x - 1) + 7$

3. Distribute and simplify: $g(x-1) = -5x^2 + 10x - 5 + 6x - 6 + 7$

4. Combine like terms: $g(x-1) = -5x^2 + 16x - 4$

Thus, $g(x-1) = -5x^2 + 16x - 4$.

Would you like further details or have any questions?

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Here are some related questions to further your understanding:

1. How would you find $g(x+1)$ for this function?
2. What happens to the graph of $g(x)$ when we find $g(x-1)$?
3. Can we generalize the process for any polynomial function $f(x)$?
4. How does substitution affect the range of $g(x)$?
5. What is the derivative of $g(x-1)$ with respect to $x$?

Tip: When substituting $x$ with $x - k$ in a function, the graph shifts horizontally by $k$ units.