Ask a New Question
Menu

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, 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, 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 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. 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.
Skip to navigation

1 answer

To determine the correct sketch for the function with an initial value of 2 and a slope of -1, we need to examine the key aspects of a linear function defined by these parameters.

  1. Initial Value: The function's initial value (or y-intercept) of 2 means that when \( x = 0 \), \( y = 2 \). This intersects the y-axis at the point (0, 2).

  2. Slope: A slope of -1 indicates that for every unit increase in \( x \), \( y \) decreases by 1. Therefore, the line will be descending from left to right.

Given these points about the function, let's analyze the provided sketches:

  • The first image features an upward line with coordinates of the x-intercept at (-1, 0) and the y-intercept at (0, 2). Since we have a downward slope, this sketch is incorrect.

  • The second image shows an upward line and does not conform to the slope of -1, hence incorrect.

  • The third image also shows an upward line with a y-intercept of (0, 2), x-intercept at (-2, 0). While it has the correct y-intercept, the slope is not negative, making it incorrect.

  • The fourth image depicts a downward line (crossing the y-axis at (0, 2) and the x-axis at (2, 0)). Since the slope here is negative, this aligns with the requirement of the slope being -1. However, the x-intercept should actually be (2, 0), which is incorrect.

Based on the given information and the characteristics of the function, there should be a downward sloping line crossing through (0, 2) and continuing downward, which would eventually cross the x-axis at (3, 0) (due to the slope of -1). It appears none of the options depict this accurately.

If we need to pick the best answer closely resembling the function characteristics, it would be the fourth sketch. Although incorrect in its intercepts, its downward slope reflects the slope of -1, consistent with the function's behavior. Thus, select the fourth image.