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In quadratic functions, the vertex of a parabola is a crucial point that tells us the peak or the lowest point of the curve, depending on its orientation. In the case of the function \( h(x) = \frac{1}{3}x^2 \), we find the vertex using the formula \( x = -\frac{b}{2a} \).

• If \( a > 0 \), the parabola opens upwards and the vertex is the minimum point.
• If \( a < 0 \), the parabola opens downwards and the vertex is the maximum point.

For this function, \( a = \frac{1}{3} \), \( b = 0 \), and \( c = 0 \). Therefore, \( x = 0 \) gives us the x-coordinate of the vertex.Plugging this back into our function, \( h(0) = \frac{1}{3}(0)^2 = 0 \), reveals the y-coordinate of the vertex, completing the vertex as \( (0, 0) \). The vertex tells us important information about the function's graph and helps us understand the function's behavior, especially in optimization problems.

The axis of symmetry of a parabola is an imaginary line that splits the parabola exactly in half, making each side a mirror image of the other. It is always a vertical line for quadratic functions in standard form \( ax^2 + bx + c \).

• For the function \( h(x) = \frac{1}{3}x^2 \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \).
• Given that \( b = 0 \), the formula simplifies to \( x = 0 \).

This line \( x = 0 \) not only serves as a useful guidepost when graphing the parabola but also indicates that the parabola is symmetric around the y-axis.Understanding the axis of symmetry is key in graphing because it aids in accurately sketching the parabola and verifying that it is symmetric. It's also insightful for problems involving symmetry in nature and design.

The standard form of a quadratic function is expressed as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Here are some crucial aspects to consider:

• \( a \) determines the direction and width of the parabola.
• \( b \) influences the position of the axis of symmetry.
• \( c \) represents the y-intercept of the graph.

For the function \( h(x) = \frac{1}{3}x^2 \), we have \( a = \frac{1}{3} \), \( b = 0 \), \( c = 0 \).The fact that \( b \) and \( c \) are both zero makes the expression \( \frac{1}{3}x^2 \) quite simple but insightful. It emphasizes the core nature of a quadratic function and demonstrates how the term \( ax^2 \) inherently forms a parabola.The standard form is foundational not only for graphing but also for deriving other forms and performing operations like completing the square. Mastering this concept enhances your ability to tackle quadratic equations with confidence.