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Part (a): Finds f(x-2). Award 4 points for the fully simplified correct quadratic x² − 9x + 17 (equivalent orderings such as −9x + x² + 17 accepted). Award 2 points if the correct substituted form (x−2)² − 5(x−2) + 3 is explicitly shown but the student makes an arithmetic or algebraic slip when expanding or combining like terms. Award 1 point if the student substitutes x−2 for x but produces a structurally incorrect expression due to missing parentheses (e.g., x−2² − 5x−2 + 3) with no valid intermediate expression. Award 0 points for no work or a fundamentally invalid method. Errors in this part do not carry forward to other parts.

Student explicitly shows substitution (x-2)^2 - 5(x-2) + 3, expands to x^2 - 4x + 4 - 5x + 10 + 3, and correctly simplifies to x^2 - 9x + 17. Image confirms fully correct work and final answer.

Part (b): Finds 5f(x). Award 4 points for the fully simplified correct expression 5x² − 25x + 15. Award 2 points if the student writes the correct unsimplified form 5(x² − 5x + 3) but fails to distribute fully, or distributes to all but one term (e.g., 5x² − 25x + 3 or 5x² − 5x + 15). Award 0 points for no valid multiplication by 5 or an invalid expression. Errors in this part do not carry forward to other parts.

Student writes the fully simplified correct expression 5x^2 - 25x + 15 in the boxed answer for part (b). Image confirms correct final answer.

Part (c): Finds f(5x). Award 4 points for the fully simplified correct expression 25x² − 25x + 3. Award 2 points if the correct substituted form (5x)² − 5(5x) + 3 is shown but the student makes an arithmetic error in simplifying (e.g., writing 5x² instead of 25x²). Award 1 point if the student substitutes 5x for x but produces an incorrect form such as 5x² − 25x + 3, showing an attempt at substitution with a structural error. Award 0 points for no valid substitution shown. Errors in this part do not carry forward to other parts.

Student writes only the final answer 25x^2 - 25x + 15 with no visible substituted form or intermediate steps. The correct expression is 25x^2 - 25x + 3. Because no valid substitution work is shown, per rubric this must receive 0 points, though the correct first two terms suggest possible unseen correct substitution.

Part (d): Finds f(x) − f(1). Award 4 points for the fully simplified correct expression x² − 5x + 4. Award 3 points if the student correctly evaluates f(1) = −1 and sets up (x² − 5x + 3) − (−1) correctly but makes a minor arithmetic slip in the final combination. Award 2 points if the student correctly evaluates f(1) = −1 but makes a sign error when removing parentheses (e.g., x² − 5x + 3 − 1), OR if the student computes an incorrect value for f(1) but correctly simplifies (x² − 5x + 3) minus their computed f(1) with full consequential credit. Award 1 point if the student correctly evaluates f(1) = −1 but makes no further valid progress, OR if the student writes f(x) − 1 (treating f(1) as 1) and simplifies to x² − 5x + 2. Award 0 points if f(1) is not evaluated and no valid setup is shown. Consequential credit is awarded for correct algebraic simplification based on an incorrect f(1).

Student correctly evaluates f(1) = 1 - 5 + 3 = -1 and sets up f(x) - f(1) = f(x) + 1, which is algebraically correct. However, the final simplification is x^2 - 5x + 2 instead of x^2 - 5x + 4, indicating a minor arithmetic slip when combining the constant terms (3 + 1 = 2 rather than 4). Image confirms correct setup and final arithmetic error.