Final answer:
The graph of f(x) transforms to g(x) by changing the slope from -2 to 1/3, changing the y-intercept from -1 to +5, and reversing the direction of the slope. This results in the line sloping upwards and a vertical translation of 6 units up.
Step-by-step explanation:
The transformation from the graph of f(x)= -2x -1 to g(x)= 1/3x + 5 involves several steps.
- Change in slope from -2 to 1/3. The original graph has a slope of -2, which means for every increase of 1 on the x-axis, the y-value decreases by 2. In the transformation, the slope changes to 1/3, meaning for every increase of 1 on the x-axis, the y-value now increases by 1/3.
- Change in y-intercept from -1 to +5. The original graph intersects the y-axis at -1, and after the transformation, it intersects the y-axis at +5.
- Change in the direction of the line. The original line slopes downwards from left to right due to the negative slope. After transforming the slope to positive, the line slopes upwards from left to right.
- Vertical translation of 6 units up. This is due to the y-intercept moving from -1 to +5.
These transformations significantly alter the appearance and position of the line on a graph.