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:

Define the system of equations

$$
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$$

Describe the graphical components

• A downward-opening parabola with vertex at \((0, 4)\) and \(y\)-intercept at \((0, 4)\).
• A straight line with a slope of \(1\) and \(y\)-intercept at \((0, 2)\).

Determine the solution method from the graph

• Find the points of intersection of the parabola and the line.
• The \(x\)-coordinates of these intersection points, which are \(x = -2\) and \(x = 1\), are the solutions to the equation.

Answer:

To solve the equation \(-x^2 + 4 = x + 2\) using a system of equations, you would graph the following two equations:

1. A quadratic function (parabola): \(y = -x^2 + 4\)
2. A linear function (line): \(y = x + 2\)

The solutions to the equation are the \(x\)-coordinates of the points where the parabola and the line intersect. In this case, the graphs intersect at \((-2, 0)\) and \((1, 3)\), so the solutions are \(x = -2\) and \(x = 1\).