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, 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 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.

Answer 1

Given the conditions of the function:

1. Initial value (y-intercept): -1 (the line should cross the y-axis at (0, -1))
2. Slope: 3 (this means for every 1 unit increase in x, y increases by 3 units)

With a y-intercept of -1 and a positive slope of 3, we can write the equation for the line as:

\[ y = 3x - 1 \]

Using this equation, we can determine the x-intercept. To find the x-intercept, set \( y = 0 \) and solve for \( x \):

\[ 0 = 3x - 1 \ 3x = 1 \ x = \frac{1}{3} \approx 0.3 \]

From this, we can deduce that the line intersects the x-axis at approximately \( (0.3, 0) \) and the y-axis at \( (0, -1) \).

Now evaluating the response options based on these criteria:

1. First option: Line intersects the x-axis at \( (0.3, 0) \) and the y-axis at \( (0, -1) \). (Correct)
2. Second option: Line intersects the x-axis at \( (-1, 0) \) and y-axis at \( (0, 3) \). (Incorrect)
3. Third option: Line is downward sloping, with intercepts that do not match our required slope and intercepts. (Incorrect)
4. Fourth option: Line intersects the x-axis at \( (3, 0) \) and y-axis at \( (0, -1) \). (Incorrect)

Therefore, the correct response is the first option, which accurately represents the function with an initial value of -1 and a slope of 3.