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)?

To solve the equation \( -x^{2}+4=x+2 \) using a system of equations, we can rewrite it in a form that allows us to graph two separate functions. ### Step 1: Rewrite the equation We can rearrange the equation to set it to zero: \[ -x^{2} - x + 4 - 2 = 0 \] This simplifies to: \[ -x^{2} - x + 2 = 0 \] ### Step 2: Define the functions We can define two functions based on the original equation: 1. \( f(x) = -x^{2} + 4 \) (a downward-opening parabola) 2. \( g(x) = x + 2 \) (a straight line with a slope of 1) ### Step 3: Graph the functions To find the solution(s) to the equation \( -x^{2}+4=x+2 \), we will graph both functions \( f(x) \) and \( g(x) \) on the same coordinate plane. - **Graph of \( f(x) = -x^{2} + 4 \)**: - This is a parabola that opens downwards. - The vertex of the parabola is at the point \( (0, 4) \). - The x-intercepts can be found by setting \( f(x) = 0 \): \[ -x^{2} + 4 = 0 \implies x^{2} = 4 \implies x = \pm 2 \] - Thus, the parabola intersects the x-axis at \( (-2, 0) \) and \( (2, 0) \). - **Graph of \( g(x) = x + 2 \)**: - This is a straight line with a y-intercept at \( (0, 2) \) and a slope of 1. - The line will rise diagonally from left to right. ### Step 4: Finding the solution(s) The solutions to the equation \( -x^{2}+4=x+2 \) correspond to the points where the graphs of \( f(x) \) and \( g(x) \) intersect. To find these intersection points: 1. Plot both functions on the same graph. 2. Identify the points where the two graphs cross each other. ### Conclusion The x-coordinates of the intersection points will give us the solutions to the equation \( -x^{2}+4=x+2 \). We can also find the exact points of intersection algebraically by solving the system of equations: \[ -x^{2} + 4 = x + 2 \] This will yield the same solutions as the graphical method. Would you like me to solve the equation algebraically to find the exact solutions?