Determine g(x−1)g(x−1) for the following function.
 

g(x)=−5x^2+6x+7


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.

Determine g(x−1) for the following function: g(x) = −5x^2 + 6x + 7

Learn to determine g(x-1) for the function g(x) = -5x^2 + 6x + 7 by applying substitution and expanding binomials. This step-by-step solution walks through transforming quadratic functions and combining like terms, suitable for students in Grades 9-11.

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