Question

describe the graph that would be used to solve the equation (-x^2 + 4 = x + 2) using a system of equations. how will you use the graph to find the solution(s)?

Explanation:

• To set up the system, split the original equation into two separate functions equal to $y$: one is the quadratic function $y = -x^2 + 4$ (a downward-opening parabola with vertex at $(0,4)$), and the other is the linear function $y = x + 2$ (a straight line with slope 1 and y-intercept 2).
• When the two graphs intersect, their $y$-values (and thus the expressions) are equal, so the $x$-values of these intersection points satisfy the original equation $-x^2 + 4 = x + 2$. These $x$-values are the solutions to the equation.

Answer:

1. The graph consists of the downward-opening parabola $y = -x^2 + 4$ and the straight line $y = x + 2$.
2. The solutions are the $x$-coordinates of the intersection points of the two graphs.