If f(x) = 2x² - 3x + 5, what is f(-2)?

The function value f(-2) equals 19 when the input -2 is substituted for every instance of x in the expression 2x² - 3x + 5, yielding 2(-2)² - 3(-2) + 5 = 2(4) + 6 + 5 = 8 + 6 + 5 = 19 after carefully respecting the order of operations and sign management during evaluation.

A. -9

This value would result from multiple sign errors during substitution: executing 2(-2)² - 3(-2) + 5 as 2(-4) - (-6) + 5 = -8 + 6 + 5 = 3 (not -9), or more plausibly calculating 2(-2)² as 2(-4) = -8 (incorrectly applying the negative sign inside the square), then -3(-2) as -6 (missing the double negative), yielding -8 - 6 + 5 = -9. This error pattern reveals two critical misconceptions: (1) failing to recognize that exponentiation precedes sign application in -2² versus (-2)² though parentheses in f(-2) mandate (-2)² = 4, students might mentally drop parentheses and compute -(2²) = -4; (2) mishandling the double negative in -3(-2), treating it as -6 rather than +6. The compound sign errors produce a negative result despite all terms contributing positively when evaluated correctly a dramatic outcome reversal highlighting how sign management deficiencies fundamentally distort function evaluation. This distractor effectively identifies learners with weak order of operations understanding combined with sign manipulation difficulties, a dangerous combination requiring explicit reinforcement of exponent precedence (parentheses first, then exponents) and negative multiplication rules through concrete number line demonstrations showing direction changes during multiplication.

B. 3

This value emerges from partially correct sign handling but incomplete evaluation: computing 2(-2)² = 2(4) = 8 correctly, then -3(-2) = +6 correctly, but then executing 8 + 6 - 5 = 9 (not 3) or 8 - 6 + 5 = 7 neither yielding 3. More plausibly, students executed 2(-2)² = 2(4) = 8 correctly but then miscalculated -3(-2) as -6 (single negative error), yielding 8 - 6 + 5 = 7, then adjusted downward to 3 through unstructured correction. Another pathway involves solving 2x² - 3x = 5 for x = -2 (treating the expression as an equation) rather than evaluating the function. The value 3 also equals f(1) = 2 - 3 + 5 = 4 (not 3) suggesting possible input substitution error where students evaluated f(1) instead of f(-2) through sign omission during input reading. This distractor proves particularly instructive for diagnosing whether students distinguish between function evaluation (substituting input into expression) versus equation solving (finding inputs yielding specific outputs) a foundational conceptual distinction where procedural similarity (both involve expressions with variables) masks fundamentally different objectives requiring distinct solution approaches.

C. 15

This amount falls 4 units short of the correct value, suggesting students likely computed 2(-2)² = 8 and -3(-2) = 6 correctly but then omitted the constant term +5 entirely or executed 8 + 6 + 1 = 15 through digit substitution. Another plausible error pathway involves calculating 2(-2)² = 2(4) = 8 correctly but then executing -3(-2) as +3 (halving the coefficient) rather than +6, yielding 8 + 3 + 5 = 16 then adjusting downward to 15. Students might recognize the need for three terms but miscalculate one component then approximate to the nearest choice. The 4-unit deficit corresponds precisely to half the middle term's contribution (6 ÷ 1.5 = 4) revealing possible coefficient misreading (3 as 1.5) or arithmetic error during multiplication. This distractor effectively targets learners with partial evaluation competence who execute some substitution steps correctly but introduce arithmetic errors or term omissions during multi-step calculations a transitional deficiency requiring explicit term-by-term evaluation protocols with intermediate result recording to prevent cognitive overload and transcription errors during sequential computation.

D. 19

This value correctly represents f(-2) through meticulous substitution and evaluation with comprehensive verification. Step-by-step evaluation: substitute x = -2 → 2(-2)² - 3(-2) + 5; evaluate exponent first → 2(4) - 3(-2) + 5; execute multiplications → 8 - (-6) + 5; simplify double negative → 8 + 6 + 5; sum sequentially → 14 + 5 = 19. Alternative pathway: recognize quadratic symmetry f(-2) = 2(4) + 6 + 5 = 8 + 6 + 5 = 19. Calculator verification: entering 2*(-2)^2 - 3*(-2) + 5 yields 19. Table verification: constructing input-output table with x = -2 confirms output 19. Graphical interpretation: the parabola y = 2x² - 3x + 5 passes through point (-2, 19) when plotted. This solution demonstrates comprehensive mastery of function evaluation including precise substitution respecting parentheses, order of operations adherence (exponents before multiplication before addition), sign management during negative input processing (particularly double negatives yielding positive contributions), and multi-step computation sequencing without term omission integrated competencies essential for calculus (limit evaluation), physics (position function analysis), economics (cost/revenue function interpretation), and data science (model prediction) where accurate function evaluation forms the foundation for increasingly sophisticated quantitative modeling and analysis across disciplines.

Conclusion

The function value 19 emerges through rigorous substitution of x = -2 into the quadratic expression with meticulous attention to order of operations, sign management (particularly the double negative yielding +6), and complete term inclusion without omission. This problem reinforces critical algebraic evaluation competencies essential across mathematical domains: respecting parentheses during substitution to preserve intended operations, executing exponentiation before multiplication in the order of operations hierarchy, managing sign changes during negative input processing (especially recognizing -3 × -2 = +6), and maintaining computational integrity through multi-step evaluation without term omission or arithmetic error. Mastery of these integrated skills proves indispensable for calculus (function limit determination), differential equations (solution verification), optimization (objective function evaluation), and machine learning (activation function computation) where precise function evaluation determines solution validity and model behavior prediction. The distractors strategically target pervasive error patterns including exponent-sign confusion, double-negative mishandling, and term omission highlighting the necessity of explicit evaluation protocols with intermediate result recording and mandatory verification through alternative computational pathways to catch sign errors and arithmetic slips that fundamentally distort output values with potentially severe consequences in applied modeling contexts where function outputs drive decision-making processes.