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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. 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, 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. 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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1 answer

To identify the correct sketch of the graph for the described function, we need to consider the given characteristics:

  1. Initial value of 2: This means that when \( x = 0 \), the function's output \( f(0) = 2 \). This corresponds to the y-intercept of the line where it crosses the y-axis at the point (0, 2).

  2. Slope of -1: A slope of -1 indicates that for every increase of 1 in \( x \), the value of \( y \) decreases by 1. This means that the line will be decreasing (slanting downwards) as it moves from left to right.

Given these two key features, we can narrow down our options:

  • The line must intersect the y-axis at (0, 2).
  • The line must have a downward slope, meaning it must pass through the second quadrant and extend down to the fourth quadrant.

Now let’s analyze the provided sketches:

  1. First option: A downward line with a y-intercept at (0, 2). This matches our criteria.

  2. Second option: An upward line that does not match the slope of -1 and doesn’t intersect at (0, 2).

  3. Third option: Another upward line with incorrect intersections.

  4. Fourth option: An upward line, which also fails to match the slope and intercept.

Thus, the correct sketch is the first option, which is a downward line passing through (0, 2).