Domain: all real numbers, or (-infinity, infinity)
Range: (-infinity, 9]
Maximum: 9; there is no minimum
y-intercept: (0,8)
Interval of increase: (-infinity, 1)
Interval of decrease: (1, infinity)
Average rate of change: 3 or -3
Explanations
Vertex: this is the maximum or minimum point on the graph, which is (1,9).
Domain: this is everywhere that the function is defined. Since there are no values of x for which -x^2 + 2x + 8 is not defined, the domain is all real numbers, or [tex](-\infty, \infty)[/tex].
Range: this is all y-values the function takes. We can see from the graph that the function takes on all values less than or equal to 9, so the range is (-infinity, 9].
Maximum or minimum: the maximum value is 9. There is no minimum value.
y-intercept: this is the point where the graph intersects the y-axis, which is (0,8) or simply 8. You can think of this as the value of f(x) when x is zero.
Interval of increase: f(x) is increasing on (-infinity, 1).
Interval of decrease: f(x) is decreasing on (1, infinity).
Average rate of change: the question is asking us to find the slope between the vertex and one of the x-intercepts – I will use (-2,0). The vertex is (1,9). The average rate of change is the slope between these points, which is (9-0)/(1-(-2)), or 3. If you instead used the x-intercept of (4,0), you would get an answer of -3.
