Identifying an Increasing Function: Graph Intersection at the Origin

To solve this problem, we need to identify which graph fulfills two criteria: intersecting the origin and having a positive slope (i.e., being an increasing function).

Let's examine the provided graphs:

• Criterion 1: Intersects the Origin
  A graph that intersects the origin will pass through the point $(0,0)$. This means that when $x = 0$, $y$ should also be $0$.

• Criterion 2: Increasing Function
  An increasing function is indicated by a line that has a positive slope. This means that as $x$ increases, $y$ should also increase.

Analysis of Graphs:

• The Green Graph: This graph passes through the point (0,0) but moves from the top left to the bottom right, which represents a negative slope.

• The Blue Graph: This graph also does not pass through the origin; it intersects the y-axis above the origin point.

• The Yellow Graph: This graph intersects below the origin and slants negatively, indicating a negative slope.

• The Red Graph: This graph passes through the point (0,0) and moves from the bottom left to the top right, which confirms a positive slope. Therefore, it is an increasing function that intersects the origin.

Based on the analysis above, the graph that represents an increasing function that intersects the origin is confidently identified as the red graph.

Therefore, the correct choice is $\text{the red graph}$.