Final answer:
To graph the ordered pairs and describe the pattern, plot the points on a coordinate plane and connect them. The function is nonlinear, as the graph forms a curved line. To write an equation, analyze the pattern and observe that the second differences are not constant, indicating a quadratic function. Solve a system of equations to find the values of a and b, then write the equation in the form y = ax^2 + bx + c.
Step-by-step explanation:
To graph the ordered pairs (0,-1), (1,0), (2,3), (3,8), and (4,15), plot each point on a coordinate plane. Connect the points to observe the pattern. In this case, the graph shows a curved line, indicating that the function is nonlinear.
To write an equation that represents the function, we can analyze the pattern and observe that the difference in y-values is increasing. We can use the second differences to determine the degree of the function. The second differences are 3, 5, 7, which are not constant, indicating a quadratic function.
Let's start by forming a quadratic equation in the form y = ax^2 + bx + c. Substituting the x and y values from one of the points, (0, -1), we have: -1 = a(0)^2 + b(0) + c, which simplifies to -1 = c.
Next, let's substitute the x and y values from another point, (1, 0), into the equation: 0 = a(1)^2 + b(1) - 1. Simplifying gives us a + b - 1 = 0.
Finally, let's substitute the x and y values from a third point, (2, 3), into the equation: 3 = a(2)^2 + b(2) - 1. Simplifying gives us 4a + 2b - 1 = 3.
We now have a system of equations. Solving the system (either by substitution, elimination, or matrices) will give us the values for a and b. Once we have those values, we can write the equation in the form y = ax^2 + bx + c.