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)
Responses

A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, fourth, and third quadrants. The line passes through left parenthesis 2 comma 3 right parenthesis, and left parenthesis negative 2 comma negative 5 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, fourth, and third quadrants. The line passes through left parenthesis 2 comma 3 right parenthesis, and left parenthesis negative 2 comma negative 5 right parenthesis.

A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows at both ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows at both ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.

A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 1 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 1 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.

A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows at both ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 2 comma 0 right parenthesis, and the y-axis at left parenthesis 0 comma 2 right parenthesis.
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Answer 1

To identify the correct sketch of the graph based on the features given for the function, let's break down the information:

1. **Initial Value of 2**: This indicates that when \( x = 0 \), \( y = 2 \). Therefore, the y-intercept is at (0, 2).

2. **Slope of -1**: A slope of -1 means that for every unit increase in \( x \), \( y \) decreases by 1. This produces a line that is decreasing from left to right.

With these two pieces of information, we can outline the characteristics of the function:

- The line should intersect the y-axis at (0, 2).
- Since the slope is -1, the line will move downward at a 45-degree angle as it moves from left to right.

Now, let's evaluate the given responses:

1. **Response 1**: This describes an upward line that is not consistent with a negative slope.

2. **Response 2**: This describes a downward line that intersects the y-axis at (0, 2) and has a negative slope. This is a valid representation of the function.

3. **Response 3**: This describes an upward line, which does not match with the negative slope.

4. **Response 4**: This describes another upward line, which again does not fit the requirements.

Based on the analysis, **Response 2** is the correct sketch of the graph, as it exhibits the qualitative features of the function with a downward line intersecting the y-axis at (0, 2) and a slope of -1.