A quadratic function passes through the points (1, 4), (2, 3), and (4, 15). What is the y-coefficient of the x-term in its standard form?

A quadratic function passes through the points (1, 4), (2, 3), and (4, 15). What is the y-coefficient of the x-term in its standard form?

["Solving for the y-coefficient of the x-term in a Quadratic Function Through Three Given Points", "When analyzing real-world or mathematical relationships involving curvature, quadratic functions are widely used due to their distinct U-shaped graphs. One common problem is finding the equation of a quadratic function passing through specific points—let’s explore how to determine the coefficient of the ( x )-term in standard form, given that the quadratic passes through ( (1, 4) ), ( (2, 3) ), and ( (4, 15) ).", "A quadratic function in standard form is written as:", "[\nf(x) = ax^2 + bx + c\n]", "where ( a ), ( b ), and ( c ) are constants, and we need to find the value of ( b )—the coefficient of the linear (x-term)—using the three given points.", "---", "### Step 1: Set up equations using the point values", "Substitute each point into ( f(x) = ax^2 + bx + c ):", "- For ( (1, 4) ):\n [\n a(1)^2 + b(1) + c = 4 \Rightarrow a + b + c = 4 \quad \ ext{(Equation 1)}\n ]", "- For ( (2, 3) ):\n [\n a(2)^2 + b(2) + c = 3 \Rightarrow 4a + 2b + c = 3 \quad \ ext{(Equation 2)}\n ]", "- For ( (4, 15) ):\n [\n a(4)^2 + b(4) + c = 15 \Rightarrow 16a + 4b + c = 15 \quad \ ext{(Equation 3)}\n ]", "---", "### Step 2: Solve the system of equations", "We now solve the system:", "1. ( a + b + c = 4 )\n2. ( 4a + 2b + c = 3 )\n3. ( 16a + 4b + c = 15 )", "Step 2.1: Eliminate ( c )", "Subtract Equation 1 from Equation 2:", "[\n(4a + 2b + c) - (a + b + c) = 3 - 4 \Rightarrow 3a + b = -1 \quad \ ext{(Equation 4)}\n]", "Subtract Equation 2 from Equation 3:", "[\n(16a + 4b + c) - (4a + 2b + c) = 15 - 3 \Rightarrow 12a + 2b = 12 \Rightarrow 6a + b = 6 \quad \ ext{(Equation 5)}\n]", "---", "### Step 3: Solve for ( a ) and ( b )", "Subtract Equation 4 from Equation 5:", "[\n(6a + b) - (3a + b) = 6 - (-1) \Rightarrow 3a = 7 \Rightarrow a = \frac{7}{3}\n]", "Now substitute ( a = \frac{7}{3} ) into Equation 4:", "[\n3\left(\frac{7}{3}\right) + b = -1 \Rightarrow 7 + b = -1 \Rightarrow b = -8\n]", "---", "### Step 4: Verification (optional but recommended)", "Substitute ( a = \frac{7}{3} ), ( b = -8 ) into Equation 1 to find ( c ):", "[\n\frac{7}{3} - 8 + c = 4 \Rightarrow c = 4 + 8 - \frac{7}{3} = 12 - \frac{7}{3} = \frac{36 - 7}{3} = \frac{29}{3}\n]", "Now double-check Equation 2:", "[\n4\left(\frac{7}{3}\right) + 2(-8) + \frac{29}{3} = \frac{28}{3} - 16 + \frac{29}{3} = \frac{57}{3} - 16 = 19 - 16 = 3 \quad \ ext{✓}\n]", "Equation 3 check:", "[\n16\left(\frac{7}{3}\right) + 4(-8) + \frac{29}{3} = \frac{112}{3} - 32 + \frac{29}{3} = \frac{141}{3} - 32 = 47 - 32 = 15 \quad \ ext{✓}\n]", "All equations are satisfied.", "---", "### Final Answer", "The y-coefficient of the x-term (i.e., the coefficient ( b )) in the standard form of the quadratic function is:", "[\n\boxed{-8}\n]"]

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Alternatively, accept decimal? But not natural.Wait — maybe check: is 210 possible?But since number of terms must be integer, and sum not integer at integer n, likely problem error.But for sake of format, accept approximate:Alternatively, perhaps the sequence is different.Lets correct: adjust $ S_n = 185 $ for $ n = 10 $? But not.Reframe: perhaps the sum is 185? But given 210.Perhaps the first term is 4? But given 5.For the purpose of this task, and since it must be solvable, assume a typo and use $ S_n = 220 $, $ n = 11 $. But original states 210.Alternatively, use exact: but final answer must be number.