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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
graph 1,2,3,4?
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.

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. 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.
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1 answer

Given the information that the function has an initial value of 2 and a slope equal to -1, we can determine the characteristics of the function.

  1. The initial value of 2 means that when \(x = 0\), the function value \(f(0) = 2\). This indicates that the graph intersects the y-axis at (0, 2).
  2. The slope of -1 means that for every unit increase in \(x\), the value of \(f(x)\) decreases by 1. This indicates a downward trend.

Based on this, the line should start at (0, 2) and go downwards, meaning it will slope downwards to the right.

Now let's evaluate the sketches you described:

  • Graph 1: Upward line, intersects y-axis at (0, 2) - Incorrect (it should be downward).
  • Graph 2: Upward line, not applicable - Incorrect.
  • Graph 3: Upward line again - Incorrect.
  • Graph 4: Downward line, intersects y-axis at (0, 2) - Correct.

Therefore, the correct sketch of the graph that exhibits the qualitative features of the function is Graph 4.