5. Explain what happens to the shape of the graph of $$ f(x)=a x^{2} $$ when $$ a0 $$ (when $$ a $$ is positive).

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5. Explain what happens to the shape of the graph of $$ f(x)=a x^{2} $$ when $$ a>0 $$ (when $$ a $$ is positive).

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When $$ a > 0 $$ in the quadratic function $$ f(x) = ax^2 $$, the following characteristics of the graph can be observed:

1. **Shape of the Graph**: The graph of the function is a parabola that opens upwards. This means that as $$ x $$ moves away from the vertex in both the positive and negative directions, the value of $$ f(x) $$ increases.

2. **Vertex**: The vertex of the parabola is located at the point (0, 0) when the function is in its simplest form $$ f(x) = ax^2 $$. This point is the minimum point of the graph.

3. **Y-Intercept**: The y-intercept of the graph is at the origin (0, 0). This is where the graph crosses the y-axis.

4. **Symmetry**: The graph is symmetric about the y-axis. This means that for every point $$ (x, f(x)) $$ on the graph, there is a corresponding point $$ (-x, f(-x)) $$.

5. **Effect of Changing $$ a $$**: 
   - As the value of $$ a $$ increases (still remaining positive), the parabola becomes narrower. This means that the graph rises more steeply as you move away from the vertex.
   - Conversely, if $$ a $$ is a small positive number, the parabola will be wider, indicating that it rises more gradually.

In summary, when $$ a > 0 $$, the graph of $$ f(x) = ax^2 $$ is an upward-opening parabola with a minimum point at the origin, symmetric about the y-axis, and its width is determined by the value of $$ a $$.
When $$ a > 0 $$ in $$ f(x) = ax^2 $$, the graph is an upward-opening parabola with its lowest point at the origin. It is symmetric about the y-axis, and the value of $$ a $$ affects how wide or narrow the parabola is.

Explain what happens to the shape of the graph of when (when is
positive).

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Explain what happens to the shape of the graph of when (when is positive).