Graphing and Solving Systems of Linear and Quadratic Equations

Question

In a European castle, a room with an arched ceiling is covered by a slanted roof. The ceiling is modeled by the equation:
f(x)=-x^2 +4 and the roofline by the equation: f(x)=-2x +5. Assume that the dimensions are in meters. What are the coordinates for the point of intersection of the roof with the ceiling assuming that the vertex of the parabola is (0,4)?

Answer 1

The point of intersection will be on both graphs, so we can set them equal to each other:   -x² + 4 = -2x + 5

 

Rearranging: x² - 2x + 1 = 0 This factors into (x - 1)² = 0, so x =1

 

The value of y at x = 1 is -2(1) + 5 = 3.

The point of intersection is therefore (1,3)