Question

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) 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. 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.

Answer 1

To determine the correct sketch of the graph for the given function, we need to consider two key aspects:

1. **Initial Value**: The function has an initial value of -1. This means that when \( x = 0 \), the value of the function \( f(0) = -1 \). Therefore, the y-intercept of the line should be at the point (0, -1).

2. **Slope**: The slope of the function is given as 3. A slope of 3 means that for every unit increase in \( x \), the value of \( y \) increases by 3. Therefore, the line will be rising (upward sloping).

Based on these characteristics:

- The y-intercept should be at (0, -1).
- The slope should indicate that the line rises as you move to the right.

Now, looking at the descriptions of the sketches:

1. The first option describes a downward line, which does not match our slope of 3.
2. The second option describes an upward line intersecting the x-axis at (3, 0) and y-axis at (0, -1), which matches our criteria.
3. The third option describes an upward line but intersects the x-axis at (-1, 0) and the y-axis at (0, 3), which does not match the initial value of -1.
4. The fourth option describes an upward line, but the x-intercept doesn't align with the slope described.

Thus, the correct sketch that reflects both the initial value and slope described in the question is the **second option**, which shows an upward line with the y-intercept at (0, -1).