Question
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) A: 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. B: 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. C: 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. D: 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.
Answers
GPT-4o mini
To determine the correct sketch based on the function's characteristics, let's analyze the given information:
1. **Initial Value of 2**: This means that when \( x = 0 \), \( y = 2 \). Thus, the line should intersect the y-axis at \( (0, 2) \).
2. **Slope of -1**: A slope of -1 indicates that for each unit increase in \( x \), \( y \) decreases by 1 unit. This will result in a downward sloping line.
Now let's examine the options:
- **Option A**: This describes an upward line and does not satisfy the conditions given the slope of -1. Therefore, this option is incorrect.
- **Option B**: This describes a downward line that intersects the y-axis at \( (0, 2) \). It also states that the line intersects the x-axis at \( (2, 0) \). Given the slope of -1, when \( x = 2 \), \( y \) should be \( 0 \), which matches this description. Therefore, this option is a strong candidate.
- **Option C**: This describes an upward line, which is incorrect based on our information about the slope. Therefore, this option is incorrect.
- **Option D**: Similar to C, this describes an upward line and is incorrect based on the slope. Thus, this option is also incorrect.
From the analysis, **Option B** is the correct choice, as it correctly represents a downward sloping line with an initial value of 2 at the y-axis and intersects the x-axis at the correct position based on the slope of -1.
1. **Initial Value of 2**: This means that when \( x = 0 \), \( y = 2 \). Thus, the line should intersect the y-axis at \( (0, 2) \).
2. **Slope of -1**: A slope of -1 indicates that for each unit increase in \( x \), \( y \) decreases by 1 unit. This will result in a downward sloping line.
Now let's examine the options:
- **Option A**: This describes an upward line and does not satisfy the conditions given the slope of -1. Therefore, this option is incorrect.
- **Option B**: This describes a downward line that intersects the y-axis at \( (0, 2) \). It also states that the line intersects the x-axis at \( (2, 0) \). Given the slope of -1, when \( x = 2 \), \( y \) should be \( 0 \), which matches this description. Therefore, this option is a strong candidate.
- **Option C**: This describes an upward line, which is incorrect based on our information about the slope. Therefore, this option is incorrect.
- **Option D**: Similar to C, this describes an upward line and is incorrect based on the slope. Thus, this option is also incorrect.
From the analysis, **Option B** is the correct choice, as it correctly represents a downward sloping line with an initial value of 2 at the y-axis and intersects the x-axis at the correct position based on the slope of -1.