Algebra modelling

Question

Use a graphing utility to graph the function, approximate the relative minimum or maximum of the function, and estimate the open intervals on which the function is increasing or decreasing

f(x) = -x² + 4x + 9

Find the x-value, y value of the vertex of the parabola, Find the point on the graph one unit to the right of the vertex, Find the relative minimum and/or maximum of the function. relative maximum (x,y); minimum (x,y). Find the intervals on which the function is increasing and decreasing. 

The function is increasing on which of these intervals?

Answer 1

Use this free link to graph the function and observe (https://www.desmos.com/calculator):

1. Vertex: The vertex will be the highest point of the graph: (x =2, y = 13)
2. Relative Maximum: x = 2, y = 13 (there's no relative minimum since it is downward facing)
3. The point on the graph 1 unit to the right of the vertex essentially means that for that point, x= 3 (since for vertex is at x = 2). Plugging x = 3 into the equation to get the exact value gives us: y = 12 (remember to square the x value separately, the negative value always stays!!)
4. The values on which the interval is increasing is: {-∞, 2} (all x-values before the relative maximum)
5. The values on which the interval is decreasing is: {2, ∞} (all x-values after the relative maximum)