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Chapter 8: Problem 96

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=\frac{2}{3}(x-2)^{2}-1$$

Short Answer

Expert verified
Vertex: (2, -1), Axis: x = 2, Domain: (-鈭�, 鈭�), Range: [-1, 鈭�)

Step by step solution

01

Identify the Vertex Form

The equation is in the vertex form of a parabola, which is \( y = a(x-h)^2 + k \). From the given equation, \( y = \frac{2}{3}(x-2)^2 - 1 \), we can identify the vertex \( (h, k) \) as \( (2, -1) \).
02

Determine the Vertex

The vertex of the parabola is \( (2, -1) \), obtained from the vertex form of the equation, where \( h = 2 \) and \( k = -1 \).
03

Find the Axis of Symmetry

The axis of symmetry for a parabola in vertex form is the line \( x = h \). Therefore, for this parabola, the axis of symmetry is \( x = 2 \).
04

Determine the Domain

The domain of any parabola is all real numbers, because there are no restrictions on the values that \( x \) can take. Thus, the domain is \( (-\infty, \infty) \).
05

Determine the Range

The parabola opens upwards because \( a = \frac{2}{3} > 0 \). Since the vertex is \( (2, -1) \), the lowest point on the parabola is \( y = -1 \). Therefore, the range is \( [-1, \infty) \).
06

Sketch the Parabola

Plot the vertex \( (2, -1) \) on a coordinate plane. Drawing the axis of symmetry \( x = 2 \), plot additional points using easy values of \( x \) around the vertex, such as \( x = 1 \) and \( x = 3 \). For \( x = 1 \), \( y = \frac{2}{3}(1-2)^2 - 1 = -\frac{1}{3} \), and for \( x = 3 \), \( y = \frac{2}{3}(3-2)^2 - 1 = -\frac{1}{3} \). Using symmetry, sketch the parabola passing through these points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex
The vertex of a parabola holds a special place as it is often considered the "tip" or "turning point" of this U-shaped curve. For equations in vertex form, like our example, understanding where the vertex is, is essential for graphing. The equation: \[y = \frac{2}{3}(x-2)^2 - 1\]has a vertex at \((2, -1)\), which is quite straightforward to determine from \( h, k \) based on the form \(y = a(x-h)^2 + k\).The vertex essentially tells us:
  • Where the parabola "turns" or changes direction.
  • The maximum or minimum point if the parabola opens downwards or upwards, respectively.
This point represents a significant feature of the graph, providing vital information for sketching the parabola.
Axis of Symmetry
The axis of symmetry is another crucial element of a parabola. It's like the invisible line that cuts our parabola right down the middle. For the equation in the vertex form, this line is \(x = h\). In our specific example,\[x = 2\]is the axis of symmetry.This concept helps because:
  • It always passes through the vertex and divides the parabola into two mirror-image halves.
  • Knowing it can make graphing easier, serving as a guide.
Every point on one side of this line has a symmetrical counterpart on the other side. This symmetry is not only beautiful but simplifies many calculations when working with parabolas.
Domain and Range
In mathematics, the domain and range for a parabola define which x-values and y-values the curve includes.
  • Domain: This is the set of possible x-values. For any parabola, the domain is unrestricted, meaning all real numbers \((-\infty, \infty)\) are included. This is because no matter how far left or right we go, there's always a vertical coordinate on the parabola.
  • Range: This tells us about the potential y-values. Since our parabola opens upwards (as \(a = \frac{2}{3} > 0\)), it starts from the lowest point, which is the y-coordinate of the vertex \(-1\). Therefore, the range is \([-1, \infty)\).
The domain and range describe the behavior and limitations of the parabola on the graph, important concepts for both analyzing and understanding the graph's shape and direction.

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