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. 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, 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. 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.
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Answer 1

To choose the correct sketch of the graph of the function based on the information given, we will use the two characteristics defined: the initial value (y-intercept) and the slope.

1. Initial Value: The function has an initial value of \(-1\). This means that the graph intersects the y-axis at the point (0, -1).

2. Slope: The slope is given as \(3\). This means that for every 1 unit increase in \(x\), \(y\) increases by \(3\) units. Thus, the line rises steeply as it moves from left to right.

Considering these characteristics:

• Y-Intercept: The line must intersect the y-axis at (0, -1).
• Direction of the Line: With a positive slope of 3, the line should rise as it moves from left to right.

Now let's analyze the options:

1. The first option describes a line that intersects the y-axis at (0, -1) and exhibits an upward trend. This matches both characteristics.

2. The second option shows a downward line with intersections at (3, 0) and (0, 3). This contradicts the slope and y-intercept requirements.

3. The third option indicates a line that intersects the x-axis at (-1, 0) and the y-axis at (0, 3). This does not fit our required intercept and slope.

4. The fourth option shows a line with a y-intercept at (0, -1) but intersects the x-axis at (3, 0), which does not reflect the correct slope for the given context.

Given this analysis, the first option is the correct sketch, as it properly reflects the initial value of \(-1\) and an upward slope of \(3\).