the table shows some values of a function of the form $y = ax^2 + bx + c$.

$\begin{array}{|c|c|c|c|c|c|}\hline x & 4 & 5 & 6 & 7 & 8 \\\hline y & 21 & 32 & 45 & 60 & 77 \\\hline\end{array}$

the value of $c$, the constant of the function, is

Question

the table shows some values of a function of the form $y = ax^2 + bx + c$.

$\begin{array}{|c|c|c|c|c|c|}\hline x & 4 & 5 & 6 & 7 & 8 \\\hline y & 21 & 32 & 45 & 60 & 77 \\\hline\end{array}$

the value of $c$, the constant of the function, is

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Second Differences", "Quadratic Modeling" ], "new_concepts": [ "Quadratic Parameter Solving" ], "current_concepts": [ "Second Differences", "Quadratic Modeling", "Quadratic Parameter Solving" ] } ## Set up a system of linear equations using the table values $$ \begin{cases} a(4)^2 + b(4) + c = 21 \ a(5)^2 + b(5) + c = 32 \ a(6)^2 + b(6) + c = 45 \end{cases} \implies \begin{cases} 16a + 4b + c = 21 \quad \text{--- (1)} \ 25a + 5b + c = 32 \quad \text{--- (2)} \ 36a + 6b + c = 45 \quad \text{--- (3)} \end{cases} $$ ## Solve for the coefficients by elimination $$ \begin{aligned} \text{(2) - (1)}: \quad & 9a + b = 11 \quad \text{--- (4)} \ \text{(3) - (2)}: \quad & 11a + b = 13 \quad \text{--- (5)} \ \text{(5) - (4)}: \quad & 2a = 2 \implies a = 1 \ \text{Substitute } a = 1 \text{ into (4)}: \quad & 9(1) + b = 11 \implies b = 2 \end{aligned} $$ ## Calculate the constant term c $$ \begin{aligned} 16(1) + 4(2) + c &= 21 \ 16 + 8 + c &= 21 \ 24 + c &= 21 \ c &= -3 \end{aligned} $$ The value of c, the constant of the function, is -3. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Parameter Solving" ] }

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Question

Explanation:

Set up a system of linear equations using the table values

Solve for the coefficients by elimination

Calculate the constant term c

Answer: