Math Problem Statement

Use quadratic regression to find the equation for the parabola going through these 3 points: (-1, -23), (-3, -67), and (2, 13).

Solution

To find the equation of the parabola using quadratic regression, we need to assume the general quadratic form:

y=ax2+bx+cy = ax^2 + bx + c

Given the points (−1,−23)(-1, -23), (−3,−67)(-3, -67), and (2,13)(2, 13), we substitute these points into the equation to form a system of equations:

Step 1: Set up the equations

  1. Substituting (−1,−23)(-1, -23): a(−1)2+b(−1)+c=−23a(-1)^2 + b(-1) + c = -23 a−b+c=−23(Equation 1)a - b + c = -23 \quad \text{(Equation 1)}

  2. Substituting (−3,−67)(-3, -67): a(−3)2+b(−3)+c=−67a(-3)^2 + b(-3) + c = -67 9a−3b+c=−67(Equation 2)9a - 3b + c = -67 \quad \text{(Equation 2)}

  3. Substituting (2,13)(2, 13): a(2)2+b(2)+c=13a(2)^2 + b(2) + c = 13 4a+2b+c=13(Equation 3)4a + 2b + c = 13 \quad \text{(Equation 3)}

Step 2: Solve the system of equations

The system of equations is:

a - b + c = -23 \\ 9a - 3b + c = -67 \\ 4a + 2b + c = 13 \end{cases}$$ We can solve this system using substitution or elimination. Let’s proceed with elimination: 1. Subtract Equation 1 from Equation 2: $$(9a - 3b + c) - (a - b + c) = -67 - (-23)$$ $$8a - 2b = -44$$ $$4a - b = -22 \quad \text{(Equation 4)}$$ 2. Subtract Equation 1 from Equation 3: $$(4a + 2b + c) - (a - b + c) = 13 - (-23)$$ $$3a + 3b = 36$$ $$a + b = 12 \quad \text{(Equation 5)}$$ ### Step 3: Solve Equations 4 and 5 The new system is: $$\begin{cases} 4a - b = -22 \\ a + b = 12 \end{cases}$$ Add these two equations to eliminate $$b$$: $$(4a - b) + (a + b) = -22 + 12$$ $$5a = -10$$ $$a = -2$$ ### Step 4: Substitute $$a = -2$$ into Equation 5 $$-2 + b = 12$$ $$b = 14$$ ### Step 5: Substitute $$a = -2$$ and $$b = 14$$ into Equation 1 $$-2 - 14 + c = -23$$ $$-16 + c = -23$$ $$c = -7$$ ### Step 6: Form the final equation $$y = -2x^2 + 14x - 7$$ So, the equation of the parabola is: $$\boxed{y = -2x^2 + 14x - 7}$$ Would you like more details or have any questions? ### Relative Questions 1. How do you verify if this quadratic equation passes through all the given points? 2. What is the vertex of the parabola given by this equation? 3. How can you find the axis of symmetry for this quadratic function? 4. What is the y-intercept of the equation $$y = -2x^2 + 14x - 7$$? 5. How do you determine whether the parabola opens upwards or downwards? ### Tip Always verify your solution by substituting all the given points into your final equation to ensure it is correct.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Systems of Equations
Regression Analysis

Formulas

y = ax^2 + bx + c
System of linear equations for finding a, b, c

Theorems

Solving systems of equations
Parabolas and their properties

Suitable Grade Level

Grades 10-12

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