MHB Finding the Equation of a Parabola

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To find the equation of a quadratic function that passes through the points (-2,1), (-6,1), and (2,-7), a system of equations can be set up using the standard form y = ax^2 + bx + c. The points (-2,1) and (-6,1) share the same y-value, indicating a vertical line of symmetry at x = -4. Using the vertex form y = a(x-h)^2 + k, where h is -4, allows for substitution of the known points to solve for the constants a and k. This approach effectively utilizes the symmetry of the parabola and the given coordinates to derive the quadratic equation. The discussion emphasizes the importance of recognizing patterns in the points to simplify the problem-solving process.
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Find the equation of a quadratic function whose graph contains the given points.
(-2,1), (-6,1), (2,-7)

THANK YOU
 
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Find the equation of a quadratic function whose graph contains the given points.
(-2,1), (-6,1), (2,-7)

solve the system for $$y = ax^2+bx+c$$ ...

$a(-2)^2 + b(-2) + c = 1$

$a(-6)^2 + b(-6) + c = 1$

$a(2)^2 + b(2) + c = -7$
 
One other method ... note the two coordinates with the same y-value.

a parabola's vertical line of symmetry would pass through the midpoint of those two points, i.e. $x=-4$

using the vertex form for a quadratic equation, $y=a(x-h)^2+k$, the vertex is at the coordinates $(h,k)$.

so, you know $h=-4$ and you have three sets of coordinates ... you should be able to substitute those values to determine the constants $a$ and $k$.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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