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.

Answer 1

To sketch a graph of a linear function with an initial value of -1 and a slope of 3, follow these steps:

1. **Identify the Initial Value**: The initial value, typically the y-intercept (where the line crosses the y-axis), is -1. This means the point (0, -1) is on the graph.

2. **Determine the Slope**: The slope of the function is given as 3. This means that for every unit increase in the x-direction, the y-value increases by 3 units. Mathematically, this can be expressed as:
\[
m = \frac{\Delta y}{\Delta x} = 3 \quad \text{(slope)}
\]

3. **Plot the Initial Point**: Start your graph by plotting the point (0, -1), which is your y-intercept.

4. **Use the Slope to Find Another Point**: From (0, -1), move 1 unit to the right (in the positive x-direction) and 3 units up (since the slope is 3). This will take you to the point (1, 2).

5. **Draw the Line**: Draw a straight line through the points (0, -1) and (1, 2). This line will extend infinitely in both directions.

6. **Qualitative Features**: The graph will be a straight line that rises from left to right, given the positive slope. It will continually increase as x increases.

The resulting sketch should clearly show a line going through the points (0, -1) and (1, 2), emphasizing the upward trend that represents the positive slope of 3.