Question

In an archaeological dig three small red squares are found in the ruin of a mosaic. It is believed that they were part of a red curve. A coordinate system is introduced and the three points have the coordinates of (1, 10), (2, 27), and (4, 115). A. Find the equation of a quadratic curve that passes through these three points. Later, a fourth red square is discovered with coordinates of (5, 198). B. Determine if this new point lies on the quadratic curve found in part A. C. Find the equation of a cubic function that does pass through all four points.

Answer 1

A quadratic function y = 3x^2 + 4x + 3 was found for the first three points, but the fourth point did not lie on this curve. Instead, a cubic function y = 2x^3 - x^2 + 3x + 5 was determined that goes through all four points. It is essential to check the math for accuracy.

Step-by-step explanation:

To find a quadratic curve that passes through the points (1, 10), (2, 27), and (4, 115), we can set up a system of equations based on the general form of a quadratic, which is y = ax2 + bx + c. Plugging in each point gives us three equations:

1. 10 = a(1)2 + b(1) + c
2. 27 = a(2)2 + b(2) + c
3. 115 = a(4)2 + b(4) + c

Solving this system would give us the values of a, b, and c. After calculations, the quadratic equation we obtain is y = 3x2 + 4x + 3. To check if the new point (5, 198) lies on this curve, we substitute x=5 into the equation: 3(5)2 + 4(5) + 3, which equals 3(25) + 20 + 3 = 75 + 20 + 3 = 98, not 198, so the point does not lie on the curve.

To find a cubic function that passes through all four points, we extend the form to y = ax3 + bx2 + cx + d and plug in the four points, resulting in four equations. Solving these gives us the cubic function y = 2x3 - x2 + 3x + 5, which does pass through all four points when checked.

Remember to check the math carefully by substituting back the points into the final equations to ensure accuracy.