A function has an initial value of −1

Question

and a slope equal to 3. Choose the sketch of a graph that exhibits the qualitative features of the function.(1 point)
Responses

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.
Image with alt text: 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.

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.
Image with alt text: 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.
Image with alt text: 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. 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.
Image with alt text: 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.
Skip to navigation

Answer 1

Given that the function has an initial value of -1 (which indicates the y-intercept) and a slope of 3, we can determine the characteristics of the function:

1. Y-Intercept: The y-intercept is at \( (0, -1) \).

2. Slope: A slope of 3 means that for every unit increase in \( x \), \( y \) increases by 3 units. Therefore, the line rises steeply.

Using the slope of 3, we can calculate another point. Starting from the y-intercept \( (0, -1) \):

• When \( x = 1 \), \( y \) would be \( -1 + 3(1) = 2 \) (point (1, 2)).

Analysis of the Options:

• First Option: Intersects at \( (0, -1) \) (correct) and \( (0.3, 0) \) (incorrect, as the x-axis intersection should be at x=1 or greater after calculation).

• Second Option: Intersects at \( (0, -1) \) (correct) and \( (3, 0) \) (also feasible, since after reaching the y-intercept it does cross the x-axis).

• Third Option: Incorrect y-intercept at \( (0, 3) \) which is not consistent with the given details.

• Fourth Option: Incorrect as it is a downward line which does not have a positive slope.

Conclusion:

The correct sketch that exhibits the qualitative features of the function based on the given initial value and slope is the second option, where the line intersects the x-axis at \( (3, 0) \) and the y-axis at \( (0, -1) \).