Question

\(y=\frac{1}{2}(x-3)^2+2\)

Short answer

The vertex of the quadratic equation \(y=\frac{1}{2}(x-3)^2+2\) is at point (3, 2). The axis of symmetry is \(x = 3\). And the parabola opens upwards and is wider than \( y = x^2\).

Step-by-step solution

Identify the Vertex

In the given equation \(y=\frac{1}{2}(x-3)^2+2\), the vertex (h,k) is found by identifying the values in the equation that match the form \( y = a(x - h)^2 + k \). Here, h is the value being subtracted from \(x\) inside the parentheses and k is the value being added at the end of the equation. Therefore, the vertex of this quadratic equation is at point (3, 2).

Determine the axis of symmetry

The axis of symmetry of a parabola in this form is a vertical line passing through the x-coordinate of the vertex. Therefore, the axis of symmetry for this quadratic equation is at \(x = 3\).

Analyze the parabola's opening

The value of 'a' in the equation determines whether the parabola opens upwards or downwards and the width of the parabola. Here, the value of 'a' is \(\frac{1}{2}\), which is positive. Therefore, the parabola opens upwards. As 'a' is less than 1, the parabola is wider than the standard parabola \( y = x^2\). This can be important for graphing.

Key concepts

Vertex Form

In quadratic functions, the vertex form is an extremely useful way to express equations. It is written as \( y = a(x - h)^2 + k \). This format makes it easy to identify key characteristics of a parabola's graph, particularly the vertex—the point at which the parabola either reaches a maximum or minimum value. In this expression, \( h \) and \( k \) designate the x and y coordinates of the parabola's vertex, respectively. Identifying these values is simply done by looking at what's inside the parentheses for \( h \) and at the standalone term for \( k \). For example, in the equation \( y = \frac{1}{2}(x-3)^2 + 2 \), the vertex is \( (3, 2) \). This means the vertex occurs three units to the right and two units up from the origin in the coordinate plane. Recognizing this helps in graphing and understanding the nature of the parabola.

Axis of Symmetry

The axis of symmetry is a critical feature of parabolas in the vertex form. This refers to the imaginary vertical line that divides the parabola into two symmetrical halves. Essentially, if one side of this line reflects onto the other side, both halves will look exactly the same. In the vertex form \( y = a(x-h)^2 + k \), the axis of symmetry can easily be found. It corresponds directly with the x-coordinate of the vertex, given by \( x = h \). In our previous example, with the vertex at \( (3, 2) \), the axis of symmetry is the line \( x = 3 \). When plotting the graph of the parabola, this axis serves as a reference point, ensuring that the graph on each side is a mirror image.

Parabola Orientation

The orientation of a parabola reveals which direction it opens, either upwards or downwards, and is determined by the value of \( a \) in the vertex form equation \( y = a(x - h)^2 + k \). When \( a \) is positive, the parabola opens upwards, making the vertex the lowest point. Conversely, when \( a \) is negative, the parabola opens downwards, turning the vertex into the peak. The magnitude of \( a \) also affects the parabola's width:

• If \( |a| > 1 \), the parabola is narrower.
• If \( 0 < |a| < 1 \), the parabola is wider.

In the equation \( y = \frac{1}{2}(x - 3)^2 + 2 \), \( a = \frac{1}{2} \), which is positive, so the parabola opens upwards, making it a U-shape. The value \( \frac{1}{2} \) is also less than 1, indicating that the parabola is wider than the standard \( y = x^2 \). This understanding aids in visualizing or sketching the correct shape on a graph.