Question

a function has an initial value of 2 and a slope equal to -1. choose the sketch of a graph that exhibits the qualitative features of the function. (1 point)

Explanation:

Step1: Analyze the function's equation

The function is linear (since it has a slope). The slope - intercept form of a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the initial value (y - intercept). Given that the initial value $b = 2$ and the slope $m=- 1$, the equation of the function is $y=-x + 2$.

Step2: Analyze the given graph

The given graph has a positive slope (since it is increasing from left to right) and a y - intercept of 2. But our function has a slope of - 1 (which means it should be decreasing from left to right) and a y - intercept of 2. So we need to find a graph that has a y - intercept at $(0,2)$ and a negative slope (decreasing). However, since the question is about choosing the sketch with the qualitative features (even though the given graph here has a positive slope, maybe there are other graphs not shown? Wait, the user provided this graph. Wait, maybe there was a mistake. Wait, the function has slope - 1, so the graph should be a line with y - intercept 2 and slope - 1. Let's check the y - intercept of the given graph: it crosses the y - axis at (0,2), which is correct for the initial value. But the slope of the given graph: from (0,2) to (1,3), the slope is $\frac{3 - 2}{1-0}=1$, which is positive. But our function has slope - 1. So if we assume that maybe this is a wrong graph and we need to identify the correct one. But since the user provided this graph, maybe there was a mis - upload. However, based on the qualitative features: initial value (y - intercept) is 2, slope is - 1. So the correct graph should have a y - intercept at 2 and a negative slope (going down from left to right). If we consider the given graph, it has the correct y - intercept but wrong slope. But maybe the question is to identify the graph with y - intercept 2 (since the initial value is 2) and then slope. Wait, the problem says "Choose the sketch of a graph that exhibits the qualitative features of the function". The qualitative features are: y - intercept at 2 and a slope of - 1 (decreasing). But the given graph has y - intercept 2 but increasing slope. Maybe there are other graphs, but since this is the only one shown, perhaps there is a mistake. But if we have to go with the given graph's y - intercept (since initial value is 2, which is the y - intercept), the graph shown has a y - intercept of 2, even though the slope is positive (but maybe the user made a mistake in the graph). So the graph with y - intercept 2 (crossing the y - axis at 2) is the one with the correct initial value.

Answer:

The graph with y - intercept at (0,2) (the given graph has this y - intercept, even though the slope is incorrect. But for the qualitative feature of initial value, this graph has the correct y - intercept. If we consider the slope, a correct graph would have a negative slope, but based on the given graph's y - intercept, it has the correct initial value feature).