Problem 22 - Algebra 2

Question

Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=\frac{1}{3}(x-1)^{2}+2 $$

Step-by-Step Solution

Answer

Vertex: (1, 2), Axis of Symmetry: x = 1, Opens Upwards.

Step 1: Identify the Given Form

The given quadratic function is \( y = \frac{1}{3}(x-1)^{2}+2 \). This is already in vertex form, which is \( y = a(x-h)^2 + k \).

Step 2: Confirm Vertex Form Components

In the vertex form \( y = a(x-h)^2 + k \), the vertex is \((h, k)\), and \(a\) determines the direction and width of the parabola's opening. Here, \(a = \frac{1}{3}\), \(h = 1\), and \(k = 2\).

Step 3: Identify the Vertex and Axis of Symmetry

The vertex of the function is \((1, 2)\). The axis of symmetry can be identified as \(x = h\), which means the axis of symmetry is \(x = 1\).

Step 4: Determine the Direction of Opening

Since \(a = \frac{1}{3}\) is positive, the parabola opens upwards. If \(a\) was negative, it would open downwards.

Key Concepts

Vertex Form

Understanding the vertex form of a quadratic function can be very helpful. The vertex form is written as \( y = a(x-h)^2 + k \). This representation clearly shows the vertex of the parabola as the point \((h, k)\).
One advantage of the vertex form is that it makes it easy to see the maximum or minimum point of the parabola, which is the vertex itself. For any given quadratic, you can quickly tell where this point is located on the graph.
In the quadratic function \( y = \frac{1}{3}(x-1)^{2}+2 \), the vertex is identified as \((h, k) = (1, 2)\). Therefore, the vertex is the point \((1, 2)\) on the graph. This is the point where the parabola changes direction.

Axis of Symmetry

The axis of symmetry is a vital concept when working with quadratic functions. This is the vertical line that runs through the vertex and divides the parabola into two mirror images.
In a graph, the axis of symmetry can be seen as the line over which the left and right sides of the parabola reflect symmetrically. This helps in visualizing the parabolic curve.
In mathematical terms, the axis of symmetry is given by the equation \( x = h \). Here, \(h\) is the x-coordinate of the vertex.

• For the function \( y = \frac{1}{3}(x-1)^2 + 2 \), the vertex is at \((1, 2)\).
• Thus, the axis of symmetry is \( x = 1 \).

This line tells us exactly where the symmetrical split of the parabola occurs, providing useful insights when graphing or analyzing the quadratic equation.

Parabola Direction

The direction in which a parabola opens is another fundamental aspect of quadratic functions. This direction is determined by the coefficient \(a\) in the vertex form equation \( y = a(x-h)^2 + k \).
If \(a\) is positive, the parabola opens upwards like a smile. This indicates that the vertex is the minimum point on the graph.
On the other hand, if \(a\) is negative, the parabola opens downwards like a frown. Here, the vertex represents the maximum point.

• In the function \( y = \frac{1}{3}(x-1)^{2}+2 \), \(a = \frac{1}{3}\) is positive, so the parabola opens upwards.

Recognizing the direction of the parabola aids in predicting the shape and orientation of the graph, providing a more comprehensive understanding of the quadratic function.