Math Problem Statement

Find an equation in the form y = ax^2 + bx + c for the parabola passing through the points (-3, 43), (5, 27), (-4, 63).

Solution

To solve for the equation of the parabola in the form y=ax2+bx+cy = ax^2 + bx + c that passes through the points (3,43),(5,27),(4,63)(-3, 43), (5, 27), (-4, 63), we need to find the coefficients aa, bb, and cc.

We will substitute each point into the general equation and solve the resulting system of three equations.

Step 1: Set up the system of equations

  1. For the point (3,43)(-3, 43): 43=a(3)2+b(3)+c    43=9a3b+c43 = a(-3)^2 + b(-3) + c \implies 43 = 9a - 3b + c

  2. For the point (5,27)(5, 27): 27=a(5)2+b(5)+c    27=25a+5b+c27 = a(5)^2 + b(5) + c \implies 27 = 25a + 5b + c

  3. For the point (4,63)(-4, 63): 63=a(4)2+b(4)+c    63=16a4b+c63 = a(-4)^2 + b(-4) + c \implies 63 = 16a - 4b + c

Step 2: Solve the system of equations

Now we have the following system of linear equations:

1. & \quad 9a - 3b + c = 43 \\ 2. & \quad 25a + 5b + c = 27 \\ 3. & \quad 16a - 4b + c = 63 \\ \end{aligned}$$ Let me solve this system to find $$a$$, $$b$$, and $$c$$.The solution to the system of equations is: $$a = 2, \quad b = -6, \quad c = 7$$ Thus, the equation of the parabola that passes through the points $$(-3, 43)$$, $$(5, 27)$$, and $$(-4, 63)$$ is: $$y = 2x^2 - 6x + 7$$ Would you like more details on how this was solved or any further questions? ### Related Questions: 1. How can we verify that this parabola passes through the given points? 2. What is the vertex of the parabola $$y = 2x^2 - 6x + 7$$? 3.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Systems of Equations

Formulas

y = ax^2 + bx + c
Substituting points into quadratic equations

Theorems

Solving systems of linear equations

Suitable Grade Level

Grades 9-11