Problem 28 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the parabola. $$x-2 y^{2}=0$$

Step-by-Step Solution

Answer

Graph the parabola \(x = 2y^2\) which opens rightward with vertex at the origin.

Step 1: Recognize the Equation Format

The given equation \(x - 2y^2 = 0\) represents a parabola. In this case, the equation is of the form \(x = ay^2\) where \(a = 2\), suggesting it opens horizontally.

Step 2: Rearrange the Equation

Solve for \(x\) in terms of \(y\) by rearranging the equation. Since \(x - 2y^2 = 0\), we can express it as \(x = 2y^2\).

Step 3: Understand the Standard Form

The equation \(x = 2y^2\) corresponds to a standard parabolic form \(x = ay^2\). This confirms that the parabola opens to the right because \(a = 2 > 0\).

Step 4: Identify Key Features

The vertex of the parabola is at the origin \((0,0)\). Since there is no \(y\)-shifting term, the axis of symmetry is the horizontal line \(y = 0\).

Step 5: Graph Using a Graphing Device

Using a graphing calculator or software, enter the equation \(x = 2y^2\). The graph will clearly show a parabola opening to the right with a vertex at the origin. Additionally, plot some points like \((0,0)\), \((2,1)\), \((2,-1)\), to verify the shape.

Key Concepts

Graphing Parabolas

Graphing a parabola is one of the fundamental skills in algebra and calculus. A parabola is a symmetrical, U-shaped curve on a graph. The shape can open either vertically or horizontally. When graphing, always consider the standard equation form, whether it's \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\). This equation format will guide you on which direction the parabola opens.

• If \(a > 0\) in \(y = ax^2\), the parabola opens upward (vertically).
• If \(a < 0\) in \(y = ax^2\), it opens downward.
• For \(x = ay^2\), the parabola opens right if \(a > 0\) and left if \(a < 0\).

Use a graphing device for an accurate representation. Input the equation and observe how the curve forms. Plot several points on the graph to ensure accuracy and visualize how it moves in space.

Equation Rearrangement

Rearranging equations is a key algebraic skill that helps transform equations into more familiar forms. It involves manipulating the equation through basic algebraic operations to solve for a particular variable or arrange it into a standard format. For example, with the equation \(x - 2y^2 = 0\), rearrange it to \(x = 2y^2\). This step is crucial for understanding the parabolic form of the equation.
When rearranging:

• Always aim to isolate the term with the variable you need to study or graph, usually by adding, subtracting, multiplying, or dividing.
• Check your work by plugging values back into the rearranged equation to ensure it's equivalent to the original.

Rearranging makes equations much easier to interpret and graph, especially when applying to geometric shapes like parabolas.

Vertex of a Parabola

The vertex of a parabola is a crucial feature as it is the point where the parabola changes direction. In parabolic equations, the vertex acts like the peak or the trough depending on the orientation. For the equation \(x = 2y^2\), the vertex is located at the origin \((0, 0)\). This is derived from the lack of linear terms that shift the graph horizontally or vertically.
To find the vertex of any parabola:

• In a horizontal parabola like \(x = ay^2\), the vertex will be at \((h, k)\), subject to vertical or horizontal shifts.
• In a vertical parabola \(y = ax^2\), it's at the point \((h, k)\).

The vertex tells you the minimum or maximum value of the parabola, a fundamental feature of its graph.

Axis of Symmetry

The axis of symmetry is an imaginary line that divides the parabola into two mirror images. It is a vertical or horizontal line passing through the vertex. For example, with \(x = 2y^2\), the axis of symmetry is horizontal, expressed as \(y = 0\). This is due to the equation being in the form \(x = ay^2\), where the parabola opens horizontally.
When identifying the axis of symmetry:

• For \(y = ax^2 + bx + c\), the axis is vertical and given by \(x = -\frac{b}{2a}\).
• For \(x = ay^2 + by + c\), it's horizontal and is \(y = -\frac{b}{2a}\).

Understanding the axis of symmetry is essential as it helps predict the behavior of the parabola and confirms that the graph is correctly plotted.