Problem 108 - A Graphical Approach to College Algebra

Question

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=(y-2)^{2}-1$$

Step-by-Step Solution

Answer

The parabola opens to the right with vertex \((-1, 2)\).

Step 1: Identify the Equation

The given equation is \( x = (y - 2)^2 - 1 \). This is the equation of a parabola with the form \(x = a(y - k)^2 + h\), indicating that it opens horizontally.

Step 2: Determine the Vertex

The vertex form of the horizontal parabola equation is \(x = a(y - k)^2 + h\), where \( (h, k) \) is the vertex. For \( x=(y-2)^2-1 \), we identify the vertex as \( (-1, 2) \).

Step 3: Determine the Direction

The coefficient \(a\) in \( (y - k)^2 \) determines the direction the parabola opens. Here, \(a = 1\) (positive), so the parabola opens to the right.

Step 4: Plot the Vertex

Plot the point \((-1, 2)\) as the vertex on the graph. This is the starting point for drawing the parabola.

Step 5: Choose Additional Points

Select values for \(y\), such as \(y = 0, 1, 3, 4\), and substitute them into the equation to find corresponding \(x\) values.

Step 6: Calculate and Plot Points

Calculate: - For \(y = 0\), \(x = 3\). - For \(y = 1\), \(x = 0\). - For \(y = 3\), \(x = 0\). - For \(y = 4\), \(x = 3\). Plot these points: \( (3, 0), (0, 1), (0, 3), (3, 4) \) on the graph.

Step 7: Draw the Parabola

Using the plotted points, sketch a smooth curve from the vertex through these points to create the parabola, ensuring it opens to the right.

Key Concepts

vertex form

The vertex form is a useful way to express the equation of a parabola, allowing you to easily identify the vertex. For horizontal parabolas, the equation takes the form \(x = a(y - k)^2 + h\), where \((h, k)\) represents the vertex of the parabola.
The vertex is a crucial point, as it is the point where the parabola changes direction. It's essentially the 'center' of the parabola.

• In the equation \(x = (y - 2)^2 - 1\), we have \(h = -1\) and \(k = 2\). Thus, the vertex is \((-1, 2)\).
• Recognizing the vertex form helps in determining how to shift the parabola on the coordinate plane.

Understanding the vertex form is a stepping stone to mastering the graphing of parabolas, especially when they are oriented in different directions.

horizontal parabolas

Horizontal parabolas open either to the right or to the left, unlike the more common vertical parabolas which open upward or downward.
The general form of a horizontal parabola is \(x = a(y - k)^2 + h\).
This form allows us to immediately see the direction in which the parabola opens:

• If \(a > 0\), the parabola opens to the right.
• If \(a < 0\), the parabola opens to the left.

In our exercise, since \(a = 1\), the parabola opens to the right. The orientation is crucial for understanding how a parabola interacts with other graphs or shapes on the coordinate grid. Connecting the equation to its graphical interpretation is key to fully grasping the concept of parabolas.

plotting points

Plotting points is essential for accurately drawing a parabola on a graph. Once you've identified the vertex, the next step is to choose additional points that the parabola will pass through.
Each chosen \(y\)-value will allow you to solve for an \(x\)-value, giving a coordinate to plot.

• For instance, selecting \(y = 0\), you calculate \(x = 3\), leading to the point \((3, 0)\).
• Another example is \(y = 1\), where \(x = 0\), resulting in the point \((0, 1)\).

It's best to pick several points on either side of the vertex to ensure the curve is drawn smoothly. After plotting points like \((3, 0)\), \((0, 1)\), \((0, 3)\), and \((3, 4)\), draw a curve that connects all points smoothly, ensuring continuity. Plotting multiple points gives a clear visual representation of the shape and direction of the parabola, offering insight into the function's behavior over a specific domain.