Question

Identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with vertex \((\mathbf{0}, \mathbf{0})\). $$y=\frac{1}{8}(x-3)^2+2$$

Short answer

The vertex of the parabola is at (3,2). The focus is at (3,2.5), and the directrix is \(y=1.5\). The axis of symmetry is \(x=3\). The graph of the parabola has been translated 3 units to the right and 2 units up from the standard parabola \(y=x^2\). The parabola is wider due to the coefficient \(a=1/8\).

Step-by-step solution

Identify the Vertex

The vertex form of a parabola is given by \(y=a(x-h)^2+k\). For the given equation \(y=(1/8) (x-3)^2+2\), the vertex is represented by the coordinates of h and k. Hence, by comparing the equations, the vertex of the parabola is at (3,2).

Identify the Focus

We have the formula for the focus as \(F=(h,k+1/4a)\). The value of a in the given equation is 1/8. Hence, the focus of the parabola will be calculated as \(F=(3, 2+4/8)=(3,2.5)\). The focus of the parabola is at (3,2.5).

Identify the Directrix

We calculate the directrix using the formula \(y=k-1/4a\). Substituting the given values, we get \(y=2-4/8=1.5\). Hence, the directrix of the parabola is \(y=1.5\).

Identify the Axis of Symmetry

The axis of symmetry of a parabola in vertex form is given by \(x=h\). Hence the axis of symmetry for the given parabola is \(x=3\).

Describe Transformations

The graph of this parabola is a transformation of the graph of the standard parabola \(y=x^2\). It has been translated 3 units to the right and 2 units up. The value of the coefficient \(a=1/8\) indicates a vertical stretch, making the parabola wider than the standard parabola.

Key concepts

Vertex Form

The vertex form of a parabola provides a way to express the equation of the curve in the format \( y = a(x-h)^2 + k \). In this structure, the coordinates \((h, k)\) describe the vertex of the parabola, showing the exact point where it changes direction. For the equation given, \( y = \frac{1}{8}(x-3)^2 + 2 \), we can identify the vertex by aligning it with the vertex form. Here, \( h \) is 3 and \( k \) is 2, thus making the vertex \((3, 2)\). This understanding of vertex form allows us to quickly determine the pinnacle of the parabola's curve.

Knowing the vertex helps visualize and graph the parabola efficiently, as it gives a clear starting point for plotting.

Focus of a Parabola

The focus of a parabola is a specific point related to its curvature that significantly influences the shape of the graph. This is found by using the formula \( F = (h, k + \frac{1}{4a}) \). Consider the equation \( y = \frac{1}{8}(x-3)^2 + 2 \), where "\( a \)" is \( \frac{1}{8} \). Substituting into the formula, we calculate the focus as \( F = (3, 2 + \frac{4}{8}) = (3, 2.5) \).

The focus plays a key role in determining the reflective properties of the parabola and helps in understanding how the parabola opens around it.

Directrix

The directrix of a parabola is a line that runs parallel to the vertex axis, defined using the formula \( y = k - \frac{1}{4a} \). For our example, \( y = \frac{1}{8}(x-3)^2 + 2 \), substituting the known values results in the directrix \( y = 2 - \frac{4}{8} = 1.5 \).

This line, along with the focus, dictates the set of points (parabola) such that each point is equidistant to the directrix and the focus. The directrix establishes a balance for the parabola’s opening.

Axis of Symmetry

The axis of symmetry is an essential characteristic line for any parabola, running vertically through its vertex, given by the equation \( x = h \). In the case of our equation \( y = \frac{1}{8}(x-3)^2 + 2 \), the axis aligns with \( x = 3 \).

This axis serves as a mirror line, showing where the graph of the parabola splits into two symmetry parts. Understanding this helps accurately draw and analyze its shape.

Graph Transformations

Graph transformations involve shifting, reflecting, compressing, or stretching a graph. Here, for the parabola \( y = \frac{1}{8}(x-3)^2 + 2 \), several transformations modify the standard \( y = x^2 \) graph:

• Translation: "3 units to the right" and "2 units up" shift the graph based on the vertex \((3, 2)\).
• Vertical Stretch: The coefficient \( a = \frac{1}{8} \) causes the parabola to widen compared to the narrower standard \( y = x^2 \) parabola.

These transformations illustrate how a base model can change through different movements and adjustments in positioning and width.