Final answer:
To solve f(x) = g(x), write a quadratic function with vertex (2, 5) and y-intercept 1, and a linear function that passes through (8, -3) with a given slope. Set the two equations equal and solve for x to find their intersection points.
Step-by-step explanation:
To solve the equation f(x) = g(x), where f is a quadratic function and g is a linear function, we first need to write down the equations of these functions based on the information provided.
The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Given the vertex (2, 5) for our quadratic function f, we can write the equation as f(x) = a(x - 2)^2 + 5. Since the y-intercept is 1, we can substitute x=0 into the equation to find a. So, f(0) = a(0 - 2)^2 + 5 = 1, giving a = -1. Our quadratic function is therefore f(x) = -(x - 2)^2 + 5.
The linear function g is given by g(x) = mx + b, where m is the slope and b is the y-intercept. However, the slope is not provided directly in the question. Instead, we do know that the line passes through the point (8, -3). Without the slope, we cannot proceed to find the equation of g, which suggests there might be missing information or a typo within the problem.
Assuming the slope is known, once you have the equation for g, you can solve f(x) = g(x) by setting their equations equal to each other and solving for x. This will give the points where the quadratic graph intersects the linear graph.