Problem 23 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of svmmetrv. $$ h(x)=\frac{1}{3} x^{2} $$

Step-by-Step Solution

Answer

The vertex is at (0,0) with the axis of symmetry at x=0.

Step 1: Identify the Basic Form of the Quadratic Function

The given function is \( h(x) = \frac{1}{3}x^2 \). This is in the standard form of a quadratic function \( ax^2 + bx + c \), where \( a = \frac{1}{3} \), \( b = 0 \), and \( c = 0 \).

Step 2: Find the Vertex of the Parabola

For a quadratic function in standard form \( ax^2 + bx + c \), the vertex is found at \( x = -\frac{b}{2a} \). Since \( b = 0 \), the vertex is at \( x = 0 \). Plugging \( x = 0 \) back into the function gives \( h(0) = \frac{1}{3}(0)^2 = 0 \). Thus, the vertex is at the point \((0, 0)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a parabola in the form \( ax^2 + bx + c \) is the vertical line \( x = -\frac{b}{2a} \). Since \( b = 0 \), the axis of symmetry is \( x = 0 \).

Step 4: Sketch the Graph

Draw the graph of the quadratic function \( h(x) = \frac{1}{3}x^2 \). The vertex is at \((0, 0)\), and the parabola opens upwards because \( a = \frac{1}{3} > 0 \). The axis of symmetry divides the parabola into two symmetric halves along the line \( x = 0 \). Use points like \((1, \frac{1}{3})\) and \((-1, \frac{1}{3})\) to help sketch the curve.

Key Concepts

Vertex of a Parabola

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.

Axis of Symmetry

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.

Standard Form of a Quadratic Function

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.