Problem 33 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ H(x)=\frac{1}{2}(x-6)^{2}-3 $$

Step-by-Step Solution

Answer

Vertex: (6, -3); Axis of symmetry: x = 6.

Step 1: Identify the Form of the Quadratic Function

The given quadratic function is in the vertex form, which is generally written as \( f(x) = a(x-h)^2 + k \). For the function \( H(x) = \frac{1}{2}(x-6)^2 - 3 \), we can identify \( a = \frac{1}{2} \), \( h = 6 \), and \( k = -3 \). This form is useful to extract the vertex directly.

Step 2: Find the Vertex

In the vertex form of the quadratic, \( (h, k) \) represents the vertex of the parabola. Thus, for the function \( H(x) = \frac{1}{2}(x-6)^2 - 3 \), the vertex is at the point \( (6, -3) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a quadratic function in vertex form is given by the line \( x = h \). Hence, for our function, the axis of symmetry is \( x = 6 \). This is a vertical line that passes through the vertex.

Step 4: Sketch the Graph

Graph the quadratic by plotting the vertex at \( (6, -3) \). The parabola will open upwards since \( a = \frac{1}{2} \) is positive. The axis of symmetry is the vertical line \( x = 6 \), and the vertex is the lowest point on the graph. Sketch the parabola ensuring it is symmetric about this line.

Step 5: Label the Vertex and Axis of Symmetry on the Graph

On the graph, clearly label the vertex with its coordinates \( (6, -3) \) and draw the axis of symmetry as a dashed line along \( x = 6 \). Ensure these labels are clear and easily identifiable.

Key Concepts

Vertex form

When dealing with quadratic functions, the vertex form is an extremely useful format for graphing purposes. Vertex form of a quadratic function is expressed as \( f(x) = a(x-h)^2 + k \). In this form, the key to understanding and graphing the function lies in the parameters \( h \) and \( k \), which represent the vertex's coordinates.
For our function \( H(x) = \frac{1}{2}(x-6)^2 - 3 \), it’s easy to see that the vertex is located at \( (h, k) = (6, -3) \).
This tells us exactly where the parabola reaches its minimum or maximum point on the graph. Knowing the vertex helps in accurately plotting the shape and orientation of the parabola, as this is the point where the graph changes direction.

• Vertex: Defines the parabola's highest or lowest point.
• Easily derived from the vertex form equation.

Grasping the concept of vertex form simplifies the process of drawing and interpreting quadratic graphs.

Axis of symmetry

The axis of symmetry in a quadratic function serves as a foundational guide from which the graph can be properly visualized and drawn symmetrically. In the vertex form \( f(x) = a(x-h)^2 + k \), the axis of symmetry is directly related to the \( h \) value.
It is mathematically represented by the vertical line \( x = h \).
For the specific function \( H(x) = \frac{1}{2}(x-6)^2 - 3 \), this simplifies to \( x = 6 \). Effectively, this line traverses through the vertex and divides the parabola into two congruent halves.

• It ensures the graph's mirror-like appearance on both sides of this axis.
• Is always a vertical line.
• Key for determining how the parabola will open and be arranged.

Understanding and marking the axis of symmetry significantly aids in constructing an accurate graph.

Parabola

A parabola is the symmetric curve formed by a quadratic function, recognizable by its distinct shape and properties. It opens either upward or downward, depending on the sign and value of the coefficient \( a \) in its equation.
In vertex form, \( a(x-h)^2 + k \), the parameter \( a \) dictates the parabola's direction and width.
For \( H(x) = \frac{1}{2}(x-6)^2 - 3 \), \( a = \frac{1}{2} \), meaning the parabola opens upwards and is wider than standard parabolas.
Key characteristics of a parabola include:

• Symmetric shape around its axis of symmetry.
• Vertex as the most extreme point (either the lowest or highest).
• Direction determined by the sign of \( a \), opening upwards if \( a > 0 \) or downwards if \( a < 0 \).

Drawing a parabola involves pinpointing the vertex, marking the axis of symmetry, and sketching the symmetrical curve according to its defining parameters.