Math Problem Statement
For the quadratic function f, the table shows three values of x and their corresponding values of f(x). Which equation defines f?
Solution
The question involves determining the equation of a quadratic function from the given table of values:
\hline x & f(x) \\ \hline -1 & 10 \\ 0 & 14 \\ 1 & 20 \\ \hline \end{array}$$ We are given four possible quadratic equations to choose from, and we need to determine which one fits the data. The general form of a quadratic equation is $$f(x) = ax^2 + bx + c$$. To identify the correct equation, we can compare the values for different $$x$$ from the table to each of the options. Let me check which equation fits these conditions. I'll test the equations provided by plugging in the values of $$x = -1$$, $$x = 0$$, and $$x = 1$$ into the candidate equations and seeing which one yields the correct $$f(x)$$ values. ### Equation Options: A. $$f(x) = 3x^2 + 3x + 14$$ B. $$f(x) = 5x^2 + x + 14$$ C. $$f(x) = 9x^2 - x + 14$$ D. $$f(x) = x^2 + 5x + 14$$ I will compute these now.The calculations for each equation show the following results when plugging in the given values of $$x = -1, 0, 1$$: - **Option A**: $$f(x) = 3x^2 + 3x + 14$$ gives $$f(-1) = 14$$, $$f(0) = 14$$, $$f(1) = 20$$. This does not match the given values. - **Option B**: $$f(x) = 5x^2 + x + 14$$ gives $$f(-1) = 18$$, $$f(0) = 14$$, $$f(1) = 20$$. This does not match either. - **Option C**: $$f(x) = 9x^2 - x + 14$$ gives $$f(-1) = 24$$, $$f(0) = 14$$, $$f(1) = 22$$. This also does not match. - **Option D**: $$f(x) = x^2 + 5x + 14$$ gives $$f(-1) = 10$$, $$f(0) = 14$$, $$f(1) = 20$$. This matches the table perfectly. Thus, **Option D** is the correct equation: $$f(x) = x^2 + 5x + 14$$ Would you like further details or explanations? Here are some related questions to explore: 1. What is the general method for finding a quadratic function given three points? 2. How can you verify a quadratic function using its vertex and roots? 3. What are some applications of quadratic functions in real-world scenarios? 4. How do you complete the square to convert a quadratic equation into vertex form? 5. What is the significance of the discriminant in a quadratic equation? **Tip**: Always test a function by substituting given points to ensure accuracy when selecting equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Substitution
Formulas
f(x) = ax^2 + bx + c
Theorems
Quadratic Equation Solving
Suitable Grade Level
Grades 9-10