Question

\(f(x)=\frac{1}{2}(x-1)^2\)

Short answer

The function \(f(x)=\frac{1}{2}(x-1)^2\) represents a parabola that opens upwards and is vertically stretched by a factor of 1/2. It is horizontally shifted to the right by 1 unit. The vertex is (1, 0) and the axis of symmetry is x=1.

Step-by-step solution

Identify the parameters in the function

Our function is \(f(x)=\frac{1}{2}(x-1)^2\), and it is in the form \(f(x)=a(x-h)^2+k\). Here, a=1/2, h=1, and k=0.

Understand these parameters

In this function, the factor a stretches or shrinks vertically and the term h shifts the parabola horizontally. Here, a (>0) will mean that the parabola opens upwards and the factor of 1/2 will vertically stretch the graph by a factor of 0.5. The term h stands for horizontal shift. Here, h=1 means that the graph will shift to the right by 1. Since the term k is absent, it is considered to be 0. This represents the vertical shift. It's not shifting up or down as k=0.

Describing the vertex and axis of symmetry

Here, since our h=1 and k=0, the vertex is at point (h,k) or (1,0). The axis of symmetry of a parabola is always the spring through the vertex of the parabola. Hence, here it will be x=1.

Key concepts

Vertex Form of a Parabola

The vertex form of a parabola is a powerful way to understand the features of this type of graph. It is expressed as \( f(x) = a(x - h)^2 + k \), where \(h\) and \(k\) represent the coordinates of the vertex, and \(a\) is a constant that affects the width and direction of the parabola.

In the exercise \(f(x)=\frac{1}{2}(x-1)^2\), the variables can be directly identified: \(a = \frac{1}{2}\), \(h = 1\), and \(k = 0\). Mathematically, this gives us a clear picture: the vertex of the parabola is at (1, 0), and since \(a > 0\), the parabola opens upwards. The fraction \(\frac{1}{2}\) in \(a\) means the parabola is wider than the standard \(y=x^2\) graph.

A student looking comprehensively at a quadratic equation in vertex form can easily determine not only the graph’s peak point but also predict how it will look without plotting it point by point.

Parabola Transformations

Transformations of a parabola refer to the shifts and scalings that change its position and shape. In the given exercise, \(f(x) = \frac{1}{2}(x - 1)^2\), the transformations are quite straightforward.

Firstly, the horizontal shift is determined by the value of \(h\). Here, because \(h=1\), the entire graph is shifted right by 1 unit from the origin.

Secondly, the vertical scaling is determined by the value of \(a\). In this case, the vertical stretch is caused by \(a = \frac{1}{2}\), which results in a wider parabola since it is less than 1. A value of \(a > 1\) would make the parabola narrower. If \(a\) were negative, the parabola would open downwards, a reflection across the x-axis.

Lastly, vertical shifts are controlled by \(k\), which in this example is 0, indicating no upward or downward shift. Understanding these parameters allows students to easily imagine how the parabola will look without creating a detailed graph.

Example of Transformation

For instance, if we had the function \(g(x) = \frac{1}{2}(x - 1)^2 + 3\), this parabola would be the same shape as the one in the exercise but shifted up by 3 units because \(k=3\).

Axis of Symmetry

The axis of symmetry in a parabola is a vertical line that runs through the vertex, dividing the parabola into two mirror images. It can be described by the equation \(x = h\), where \(h\) is the x-coordinate of the vertex.

For the function \(f(x) = \frac{1}{2}(x - 1)^2\), since the vertex is at (1, 0), the axis of symmetry is the line \(x = 1\). This tells us that for every point on the parabola, there is another point mirrored across the axis of symmetry with the same distance from the axis.

The axis is essential for understanding the parabola’s symmetry and for graphing. It is also a tool that can help students find the maximum or minimum value of the function since the vertex lies on this axis.

Finding the Vertex via Axis of Symmetry

For students, identifying the axis of symmetry is key in graphing parabolas and solving related problems. It simplifies finding the vertex and is instrumental in solving quadratic equations by completing the square or in vertex form.