Final answer:
To draw the graph requested by the student, plot the vertex at (2,-3) and create a V-shaped graph with the left arm decreasing towards the vertex, and the right arm increasing after the vertex, reflecting the characteristics of an absolute value function. The graph extends infinitely along the x-axis and upwards from the vertex at y=-3.
Step-by-step explanation:
The student has asked for help to draw the graph of an absolute value function with specific features: a vertex at (2,-3), increasing from (2, ∞), decreasing from (-∞, 2), a domain of all real numbers, and a range of [-3, ∞). Additionally, the end behavior indicates that as x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) also approaches positive infinity.
To create this graph, start by plotting the vertex at (2,-3). Since it is an absolute value function, the graph will have a V-shaped form. The 'increasing' and 'decreasing' parts indicate the direction of the two arms of the V. The left arm will decrease towards the vertex as you move to the right, and then the right arm will increase after the vertex, as x becomes larger. This is consistent with the vertex form of an absolute value function, which can be written as f(x) = a|x-h| + k, where (h, k) is the vertex. Here, a will be positive because as x increases, f(x) also increases.
The domain being all real numbers means the graph will stretch infinitely along the x-axis. The range starting at -3 and going to infinity means the vertex at -3 is the lowest point, and the graph extends upward indefinitely from there. And finally, the end behavior tells us that the graph will continue to rise in both directions as we move away from the vertex, confirming that the 'arms' of the V never turn back down.