Final answer:
To create a table of values for the function f(x)=x²+4x, calculate the values of f(x) for different values of x from -7 to 4. Find the first and second finite differences by calculating the differences between consecutive values. Analyze the signs of the finite differences to determine the intervals of increasing, decreasing, concave up, and concave down. Sketch the graph using the information from the finite differences and compare it to the graph using knowledge of quadratic functions.
Step-by-step explanation:
To create a table of values for the function f(x)=x²+4x, we substitute different values of x from -7 to 4 into the function to find the corresponding values of f(x). Here is the table:
xf(x)-77-60-5-1-40-3-1-20-110015212321432
To determine the first finite difference, we find the difference between consecutive values of f(x). To determine the second finite difference, we find the difference between consecutive values of the first finite difference. Here are the first and second finite differences:
xf(x)First DifferenceSecond Difference-77-60-7-5-1-16-4017-3-1-16-2017-111600-171556212773219643211
To determine the interval(s) over which the function is increasing, decreasing, concave up, and/or concave down, we analyze the signs of the finite differences. An increasing function has positive first finite differences, while a decreasing function has negative first finite differences. A concave up function has positive second finite differences, while a concave down function has negative second finite differences. From the table, we can see that the function is increasing from x = -6 to x = 0, decreasing from x = 1 to x = 4, concave up from x = -5 to x = -3, concave down from x = -2 to x = 1, and concave up again from x = 2 to x = 4.
To sketch the graph of the function using the information from the finite differences, we start with a horizontal line at f(x) = 0 and draw increasing or decreasing line segments based on the signs of the first finite differences. We also draw concave up or concave down curves based on the signs of the second finite differences. Finally, we connect the segments and curves to get the graph of the function. To sketch the graph of the function using our knowledge of quadratic functions, we recognize that f(x) = x²+4x is a quadratic function in the form of f(x) = ax²+bx+c. The graph of a quadratic function is a parabola. Since the coefficient of x² is positive (+1 in this case), the parabola opens upwards. We can use the vertex and the y-intercept to sketch the graph accurately.
Comparing the two graphs, we see that they are the same, which confirms the accuracy of the information obtained from the finite differences.