Short Answer 
The answer explains quadratic functions as polynomials represented by y = ax² + bx + c, highlighting their x-intercepts where they cross the x-axis. It also details how to calculate the average rate of change between intervals and how to verify points as solutions to quadratic equations by substituting x-values.
Step 1: Understand Quadratic Functions and Their X-Intercepts
A quadratic function is a polynomial function of the form y = ax² + bx + c. It can have one or more x-intercepts, which are the points where the graph crosses the x-axis. For example, the function y = x² has its x-intercept at the origin (0,0) because when x = 0, y also equals 0. However, the function y = x² + 3 does not have an x-intercept at the origin since it does not cross the x-axis.
Step 2: Analyze Average Rate of Change Between Intervals
The average rate of change of a function over an interval is calculated by taking the difference in function values divided by the difference in x-values. In this case, from x = -2 to x = 0, both functions are declining. Thus, the average rate of change is negative, as we are moving from higher to lower y-values. Specifically, when looking at this interval, the graphs of both y = x² and y = x² + 3 decrease from left to right.
Step 3: Verify Solutions of Points for Quadratic Functions
To determine if a point is a solution to the quadratic equation, substitute the x-value into the equation and see if it yields the corresponding y-value. For instance, for y = x², substituting x = 2 gives y = 4, which does not match the point (2,3), making it false. Conversely, substituting x = 2 into y = x² + 3 gives y = 7, matching the point (2,7) accurately, confirming it as a correct solution.