sketch a graph of f.  f(x)=$$ \frac14x^2 $$

To sketch the graph of the function $ f(x) = \frac{1}{4}x^2 $, follow these steps:

### Key Features of the Graph:
1. **Shape**: The function is a quadratic equation and has a parabolic shape. Since the coefficient of $ x^2 $ is positive, the parabola opens upwards.
2. **Vertex**: The vertex of the parabola is at $ (0, 0) $, since there is no linear term or constant term to shift it horizontally or vertically.
3. **Axis of Symmetry**: The axis of symmetry is the vertical line $ x = 0 $, which passes through the vertex.
4. **Width**: The coefficient $ \frac{1}{4} $ makes the parabola wider compared to $ x^2 $. The larger the denominator, the wider the parabola.

### Steps to Sketch the Graph:
- Plot the vertex at $ (0, 0) $.
- Plot a few points by substituting values of $ x $ into $ f(x) $. For example:
  - When $ x = -2 $, $ f(-2) = \frac{1}{4}(-2)^2 = 1 $.
  - When $ x = 2 $, $ f(2) = \frac{1}{4}(2)^2 = 1 $.
  - When $ x = -4 $, $ f(-4) = \frac{1}{4}(-4)^2 = 4 $.
  - When $ x = 4 $, $ f(4) = \frac{1}{4}(4)^2 = 4 $.
- Draw a smooth curve through these points to form the parabola, ensuring it opens upwards and is symmetric around the y-axis.

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### Questions for deeper understanding:
1. How does the coefficient $ \frac{1}{4} $ affect the width of the parabola compared to the standard $ x^2 $?
2. What would happen to the graph if we changed the function to $ -\frac{1}{4}x^2 $?
3. How can you determine the vertex of a parabola for a more general quadratic function?
4. What is the significance of the axis of symmetry in a quadratic graph?
5. How can you calculate the points where the parabola intersects the y-axis?

**Tip**: For any quadratic function $ ax^2 + bx + c $, the graph will always be a parabola. The sign of $ a $ determines the direction it opens, and the value of $ a $ influences how "wide" or "narrow" the parabola is.

sketch a graph of f.  f(x)=\frac{1}{4}x^2

Learn how to graph the quadratic function f(x) = (1/4)x^2. This guide covers the key features of the parabola, including its vertex at (0,0), axis of symmetry, and the effect of the 1/4 coefficient on the parabola's width. Suitable for students in Grades 9-11.

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