Problem 51 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$y=(x-2)^{2}-2$$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (2, -2).

Step 1: Identify the Equation Type

The equation given is \( y = (x-2)^2 - 2 \), which is written in the form \( y = a(x-h)^2 + k \). This is the vertex form of a parabola. Therefore, the graph of this equation is a parabola.

Step 2: Determine the Vertex of the Parabola

In the vertex form of a parabola, \( y = a(x-h)^2 + k \), the vertex is given by the point \( (h, k) \). For the equation \( y = (x-2)^2 - 2 \), \( h = 2 \) and \( k = -2 \). Hence, the vertex of the parabola is \( (2, -2) \).

Step 3: Sketch the Parabola

Using the vertex \( (2, -2) \), plot this point on the coordinate plane. This parabola opens upwards because the coefficient of \( (x-2)^2 \) is positive (specifically, 1). Since the coefficient is 1, the parabola will have a standard width. Draw a smooth curve opening upwards with the vertex at \( (2, -2) \). The axis of symmetry for the parabola is the vertical line \( x = 2 \).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is an important way to express quadratic functions and is given by the equation: \[ y = a(x-h)^2 + k \] where \(a\), \(h\), and \(k\) are constants. This form clearly shows the vertex of the parabola, which is a crucial point on its graph. Here, \(h\) and \(k\) represent the coordinates of the vertex of the parabola (\(h, k\)). By using this form, it is easy to identify the vertex directly without needing to complete the square or perform any additional algebraic manipulations.

• \(a\): controls the direction and width of the parabola (\(a > 0\) means it opens upwards, \(a < 0\) means it opens downwards).
• \(h\): is the x-coordinate of the vertex, shifting the parabola left or right.
• \(k\): is the y-coordinate of the vertex, shifting the parabola up or down.

With the vertex form, you can immediately determine both the vertex and the direction in which the parabola opens, making it simple to graph.

Finding the Vertex

Finding the vertex of a parabola is straightforward when it is expressed in vertex form. In the equation \( y = (x-2)^2 - 2 \), we can directly identify the vertex since it uses the vertex form's structure. The vertex form allows you to pick out the values of \(h\) and \(k\):

• From \((x-h)^2\), we find that \(h = 2\).
• From the constant term \(-2\), we know \(k = -2\).

Therefore, the vertex of this parabola is \((2, -2)\). This point is where the parabola changes direction and is considered the maximum or minimum point depending on whether the parabola opens downwards or upwards.

Sketching Graphs

Sketching graphs of parabolas can be an intuitive process once you have determined the vertex and understand the parabolic shape. For the equation \( y = (x-2)^2 - 2 \):

• Begin by plotting the vertex \((2, -2)\) on the coordinate plane.
• Since the coefficient of \((x-2)^2\) is positive, the parabola opens upwards. The parabola's arms will extend upwards from the vertex.
• Notice that the coefficient is 1, indicating the parabola has a standard width, so its shape will be symmetrical and relatively narrow.

Draw a smooth curve through the vertex \((2, -2)\), ensuring that the curve reflects across the vertical axis that runs through the vertex. Use additional points if necessary to understand better how wide or narrow the curve will be.

Axis of Symmetry

The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. For any parabola in vertex form \( y = a(x-h)^2 + k \), the axis of symmetry is given by the line \( x = h \). This line runs through the vertex of the parabola and provides a line of reflection for each side of it.

• In the equation \( y = (x-2)^2 - 2 \), the axis of symmetry is \( x = 2 \).
• This vertical line ensures that the parabola is symmetric, with the vertex being the lowest or highest point depending on its direction.

Understanding the axis of symmetry helps when graphing because it guarantees a balanced shape and aids in determining how the graph behaves at various points.