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

\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & 3 & 4 & 5 & 6 & 7 \\
\hline
y & -10 & -16 & -24 & -34 & -46 \\
\hline
\end{tabular}

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{tabular}{|c|c|c|c|c|c|}
\hline
x & 3 & 4 & 5 & 6 & 7 \\
\hline
y & -10 & -16 & -24 & -34 & -46 \\
\hline
\end{tabular}

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 $$ \begin{cases} a(3)^2 + b(3) + c = -10 \ a(4)^2 + b(4) + c = -16 \ a(5)^2 + b(5) + c = -24 \end{cases} \implies \begin{cases} 9a + 3b + c = -10 \quad \text{--- (1)} \ 16a + 4b + c = -16 \quad \text{--- (2)} \ 25a + 5b + c = -24 \quad \text{--- (3)} \end{cases} $$ ## Eliminate the constant term to solve for coefficients $$ \begin{aligned} (2) - (1): \quad & 7a + b = -6 \quad \text{--- (4)} \ (3) - (2): \quad & 9a + b = -8 \quad \text{--- (5)} \ (5) - (4): \quad & 2a = -2 \implies a = -1 \ \text{Substitute } a = -1 \text{ into (4)}: \quad & 7(-1) + b = -6 \implies b = 1 \end{aligned} $$ ## Solve for the constant parameter $$ \begin{aligned} 9(-1) + 3(1) + c &= -10 \ -9 + 3 + c &= -10 \ -6 + c &= -10 \ c &= -4 \end{aligned} $$ The value of c, the constant of the function, is -4. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Modeling" ] }

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QUESTION IMAGE

Question

Explanation:

Set up a system of linear equations using the table values

Eliminate the constant term to solve for coefficients

Solve for the constant parameter

Answer: