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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)
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 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.

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.
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1 answer

To determine which sketch of the graph corresponds to a function with an initial value of -1 and a slope of 3, we can analyze the given features.

  1. Initial Value: The initial value refers to the \( y \)-intercept of the function. Since the initial value is -1, the graph must intersect the y-axis at \( (0, -1) \).

  2. Slope: The slope of the line is given as 3. This means that for every 1 unit increase in \( x \), \( y \) will increase by 3 units. Therefore, the line will rise steeply as it goes from left to right.

Given these two characteristics, we need to find the graph that meets both criteria:

  • Intersects the y-axis at \( (0, -1) \)
  • Has a positive slope of 3, meaning an upward slope from left to right.

Among the responses provided, the first option correctly describes a line that:

  • Intersects the y-axis at \( (0, -1) \)
  • Has a slope that indicates it rises positively

The sketch says it intersects the x-axis at \( (3, 0) \), which is consistent with a line that starts at \( (0, -1) \) and moves upward, confirming a slope of 3.

Thus, the correct response is:

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 (3, 0) and the y-axis at (0, -1).