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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. 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 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. 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.

1 answer

To identify the correct graph based on the given information, we need to recognize that the function has:

  1. An initial value (y-intercept) of \(-1\).
  2. A slope of \(3\), which indicates the line will go upwards as we move from left to right.

Given a slope of \(3\), for every unit that \(x\) increases, \(y\) will increase by \(3\). The y-intercept of \(-1\) means that when \(x = 0\), \(y = -1\).

Now, let's analyze the sketches provided in the original question:

  1. The first option describes a downward line—this cannot be correct since the slope is positive (3).

  2. The second option describes an upward line, intersects the x-axis at \( (3, 0) \) and the y-axis at \( (0, -1) \). This matches our requirements: it has a y-intercept of \(-1\) and an upward slope.

  3. The third option describes an upward line but intersects the x-axis at \((-1, 0)\)—this does not match our criteria.

  4. The fourth option describes a high positive y-intercept and intersects at \( (0, -1) \) but not at \( (3,0) \), which isn’t correct.

Therefore, the correct choice is:

A coordinate plane with four quadrants shows 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 \( (0, -1) \).

This graph reflects the specified initial value and slope of the function appropriately.