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Functions Unit Test

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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. 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.

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.

1 answer

To find the correct sketch of the graph based on the given function details, we can use the following information:

  1. Initial Value (y-intercept): The function has an initial value of 2, which means that when \( x = 0 \), \( y = 2 \). Thus, the line will intersect the y-axis at the point (0, 2).

  2. Slope: The slope of the function is -1. This means that for every increase of 1 in \( x \), \( y \) decreases by 1. This indicates a downward sloping line.

Using the slope and the initial value, we can derive the equation of the line:

  • The slope-intercept form of the line is given by: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

Plugging in the values: \[ y = -1x + 2 \] This equation tells us that the line will intersect the y-axis at (0, 2) and will be downward-sloping.

Now let’s analyze the options:

  1. First Option: The line passes through (2, 3) and (−2, −5). This is not downward-sloping and does not match our function.

  2. Second Option: An upward line that intersects the x-axis at (−1, 0) and the y-axis at (0, 2). This is also not what we want because it is upward sloping.

  3. Third Option: A downward line that intersects the x-axis at (2, 0) and the y-axis at (0, 2). This is a downward sloping line, matches the y-intercept, but intersects the x-axis at (2, 0), which is inconsistent with our slope.

  4. Fourth Option: An upward line that intersects the x-axis at (−2, 0) and the y-axis at (0, 2). This does not match as it’s upward sloping.

Given the analysis, though the best fit (the characteristics of a downward slope and y-intercept at (0, 2)) seems to be primarily discussed, no option perfectly describes the correct line for the provided attribute of the function.

The correct choice among the given options that closely fits our slope and intercept, considering the negative slope, would be the third option. But do note that the intersection with the x-axis indicated is incorrect.

If this assumption on the provided options aligns accordingly, opt for the third one for the qualitative representation.