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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 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, 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.
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1 answer

Given the initial value of the function is -1 and the slope is 3, we can derive the equation of the line. The equation of a linear function can be written in slope-intercept form as:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept (the value of y when \( x = 0 \)).

From your description:

  • The slope \( m = 3 \) means for each increase of 1 in \( x \), \( y \) increases by 3.
  • The initial value (y-intercept) means \( b = -1 \), so the line intersects the y-axis at (0, -1).

The equation of the function is therefore:

\[ y = 3x - 1 \]

To find the x-intercept (where \( y = 0 \)):

\[ 0 = 3x - 1 \] \[ 3x = 1 \] \[ x = \frac{1}{3} \]

So the line intersects the x-axis at approximately (0.33, 0), which is very close to (0.3, 0) as mentioned.

Now we need to choose the correct sketch based on this information:

  1. The line should be upward sloping because of the positive slope (3).
  2. The line must intersect the y-axis at (0, -1).
  3. The line must intersect the x-axis at approximately (0.33, 0).

From the provided sketches, you'll want to choose the one where the line meets those criteria.

It sounds like the correct response is the first image you've described, which shows an upward line that intersects the y-axis at (0, -1) and the x-axis at approximately (0.3, 0).