A function has an initial value of −1 and a slope equal to 3. Choose the sketch of a graph that exhibits the qualitative features of the function.(1 point) A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows on both the ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma 3 right parenthesis. A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis. A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the 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 3 right parenthesis. A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 0.3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis. Values are approximate.

Question

A function has an initial value of  −1   and a slope equal to 3. Choose the sketch of a graph that exhibits the qualitative features of the function.(1 point) A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows on both the ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma 3 right parenthesis. A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis. A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the 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 3 right parenthesis. A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 0.3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis. Values are approximate.

GPT-4o mini · Answer

To determine the correct sketch of the graph for the given function, we need to consider two key aspects:

Initial Value: The function has an initial value of -1. This means that when \( x = 0 \), the value of the function \( f(0) = -1 \). Therefore, the y-intercept of the line should be at the point (0, -1).
Slope: The slope of the function is given as 3. A slope of 3 means that for every unit increase in \( x \), the value of \( y \) increases by 3. Therefore, the line will be rising (upward sloping).

Based on these characteristics:

The y-intercept should be at (0, -1).
The slope should indicate that the line rises as you move to the right.

Now, looking at the descriptions of the sketches:

The first option describes a downward line, which does not match our slope of 3.
The second option describes an upward line intersecting the x-axis at (3, 0) and y-axis at (0, -1), which matches our criteria.
The third option describes an upward line but intersects the x-axis at (-1, 0) and the y-axis at (0, 3), which does not match the initial value of -1.
The fourth option describes an upward line, but the x-intercept doesn't align with the slope described.

Thus, the correct sketch that reflects both the initial value and slope described in the question is the second option, which shows an upward line with the y-intercept at (0, -1).