Math Problem Statement
For the quadratic function above, find the second difference and the values of the coefficients a, b, and c for the function y = ax^2 + bx + c.
Solution
From the image, you have a table with values for and , where is a function of . The points provided are:
x & y \\ \hline -3 & 0 \\ -2 & 0 \\ -1 & -1 \\ 0 & 0 \\ 1 & 3 \\ 2 & 8 \\ 3 & 15 \\ \end{array}$$ You're asked to find the second difference and the values of the quadratic function coefficients $$a$$, $$b$$, and $$c$$ for the function $$y = ax^2 + bx + c$$. ### Step 1: Finding the Second Difference To find the second difference, calculate the differences between consecutive $$y$$-values. #### First differences: $$\Delta_1 y = \begin{array}{c} 0 - 0 = 0 \\ -1 - 0 = -1 \\ 0 - (-1) = 1 \\ 3 - 0 = 3 \\ 8 - 3 = 5 \\ 15 - 8 = 7 \\ \end{array}$$ Thus, the first differences are: $$0, -1, 1, 3, 5, 7$$. #### Second differences: $$\Delta_2 y = \begin{array}{c} -1 - 0 = -1 \\ 1 - (-1) = 2 \\ 3 - 1 = 2 \\ 5 - 3 = 2 \\ 7 - 5 = 2 \\ \end{array}$$ The second differences are constant at 2, confirming that this is a quadratic function. The second difference is 2. ### Step 2: Determining the Quadratic Function $$y = ax^2 + bx + c$$ The second difference being constant implies the function is quadratic. The coefficient $$a$$ is half of the second difference, so: $$a = \frac{2}{2} = 1$$ Now, substitute the known $$x$$ and $$y$$ values into the quadratic function $$y = ax^2 + bx + c$$ to form a system of equations to find $$b$$ and $$c$$. Using the points $$(-3, 0)$$, $$(0, 0)$$, and $$(1, 3)$$: 1. For $$x = -3$$, $$y = 0$$: $$0 = 1(-3)^2 + b(-3) + c \quad \Rightarrow \quad 0 = 9 - 3b + c$$ $$9 - 3b + c = 0 \quad \Rightarrow \quad 3b - c = 9 \quad \text{(Equation 1)}$$ 2. For $$x = 0$$, $$y = 0$$: $$0 = 1(0)^2 + b(0) + c \quad \Rightarrow \quad 0 = c$$ So, $$c = 0$$. 3. For $$x = 1$$, $$y = 3$$: $$3 = 1(1)^2 + b(1) + 0 \quad \Rightarrow \quad 3 = 1 + b$$ $$b = 2$$ ### Step 3: Final Function Thus, the quadratic function is: $$y = x^2 + 2x$$ ### Final Answers: - Second difference: 2 - $$a = 1$$ - $$b = 2$$ - $$c = 0$$ Let me know if you'd like more details or if you have any questions! Here are five follow-up questions for practice: 1. How do you find the first difference of a sequence? 2. Why does a constant second difference imply a quadratic function? 3. What happens if the second difference is not constant? 4. How can you use other points to verify the values of $$a$$, $$b$$, and $$c$$? 5. How does changing $$a$$ in the quadratic equation affect the graph? **Tip:** The second difference method works only for quadratic sequences! If the second difference is not constant, the function is not quadratic.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Second Differences
Algebra
Formulas
y = ax^2 + bx + c
Second difference: Δ2y = Δ1y2 - Δ1y1
Coefficient of a quadratic function: a = (second difference) / 2
Theorems
Properties of Quadratic Functions
Suitable Grade Level
Grades 8-10
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