Math Problem Statement
Find the quadratic function that is the best fit for f(x) defined by the table below. x 0 2 4 6 8 10
f(x) 0 398 1603 3604 6401 9999 Question content area bottom Part 1 The quadratic function is yequalsnothing x squared plus nothing x plus left parenthesis nothing right parenthesis .
Solution
To find the quadratic function that best fits the given data, we need to fit a quadratic curve to the points. A quadratic function generally has the form:
where , , and are constants that we need to determine.
The given data points are:
\hline x & f(x) \\ \hline 0 & 0 \\ 2 & 398 \\ 4 & 1603 \\ 6 & 3604 \\ 8 & 6401 \\ 10 & 9999 \\ \hline \end{array}$$ ### Step 1: Set up a system of equations We can use the method of least squares or solve directly by using three data points to create a system of equations. We'll start by using the first three points $$(0, 0)$$, $$(2, 398)$$, and $$(4, 1603)$$, to find the coefficients $$a$$, $$b$$, and $$c$$. From the first point $$(0, 0)$$: $$f(0) = a(0)^2 + b(0) + c = 0 \quad \Rightarrow \quad c = 0$$ Now we know that $$c = 0$$, so the quadratic function simplifies to: $$f(x) = ax^2 + bx$$ ### Step 2: Use the other points to solve for $$a$$ and $$b$$ Using the second point $$(2, 398)$$: $$f(2) = a(2)^2 + b(2) = 398 \quad \Rightarrow \quad 4a + 2b = 398 \quad \text{(Equation 1)}$$ Using the third point $$(4, 1603)$$: $$f(4) = a(4)^2 + b(4) = 1603 \quad \Rightarrow \quad 16a + 4b = 1603 \quad \text{(Equation 2)}$$ ### Step 3: Solve the system of equations We now have the system of two equations: 1. $$4a + 2b = 398$$ 2. $$16a + 4b = 1603$$ We can solve this system using substitution or elimination. Let's use elimination. Multiply Equation 1 by 2: $$8a + 4b = 796 \quad \text{(Equation 3)}$$ Now subtract Equation 2 from Equation 3: $$(8a + 4b) - (16a + 4b) = 796 - 1603$$ $$-8a = -807 \quad \Rightarrow \quad a = \frac{807}{8} = 100.875$$ ### Step 4: Solve for $$b$$ Substitute $$a = 100.875$$ into Equation 1: $$4(100.875) + 2b = 398$$ $$403.5 + 2b = 398$$ $$2b = 398 - 403.5 = -5.5$$ $$b = \frac{-5.5}{2} = -2.75$$ ### Step 5: Write the final quadratic equation Now that we have $$a = 100.875$$, $$b = -2.75$$, and $$c = 0$$, the quadratic function that best fits the data is: $$f(x) = 100.875x^2 - 2.75x$$ --- Would you like more details on how to interpret the results or explore other methods of finding the best fit quadratic function? ### Follow-up Questions: 1. How do you use the least squares method to fit a quadratic function? 2. What are the benefits of using a quadratic model over linear or other models? 3. How do you check the accuracy of the quadratic fit using the full dataset? 4. What would the quadratic function look like if we added more data points? 5. How do you interpret the coefficients $$a$$, $$b$$, and $$c$$ in terms of the problem? ### Tip: To improve the fit with more data points, you could use tools like Excel or Python's NumPy to compute the coefficients using regression.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Least Squares Method
Formulas
f(x) = ax^2 + bx + c
Theorems
Method of Least Squares
Suitable Grade Level
Grades 9-11
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