Problem 5 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the parabola. $$ 4 x^{2}=-2 y $$

Step-by-Step Solution

Answer

The parabola is a downward opening parabola with vertex at (0,0).

Step 1: Rewrite the equation

First, we want to rewrite the given equation, \( 4x^2 = -2y \), in a standard form of a parabola. Start by isolating \( y \). Divide both sides of the equation by \(-2\): \[ y = -2x^2\]This gives us the equation in standard form \( y = ax^2 \) with \( a = -2 \).

Step 2: Identify the vertex

The standard form \( y = ax^2 \) indicates that the vertex of the parabola is at the origin \((0, 0)\) because the equation does not have any \(x\) or \(y\) translations.

Step 3: Determine the direction of the parabola

In the equation \( y = -2x^2 \), the coefficient of \( x^2 \) is \(-2\), which is negative. This means the parabola opens downward.

Step 4: Calculate the axis of symmetry

For the graph \( y = ax^2 \), the axis of symmetry is the vertical line that passes through the vertex. Thus, the axis of symmetry is \( x = 0 \).

Step 5: Plot key points and sketch the graph

Choose a few values of \( x \), substitute them into the equation \( y = -2x^2 \), and plot them to help sketch the graph. For example, if \( x = 1 \), then \( y = -2(1)^2 = -2 \). If \( x = -1 \), then \( y = -2(-1)^2 = -2 \). These points, along with the symmetry at \( x = 0 \), will help form the graph. The parabola is symmetric about the y-axis and opens downward with the vertex at \((0,0)\).

Key Concepts

Vertex of a Parabola

A parabola is defined mathematically and visually by its vertex, which is the highest or lowest point on its graph. The vertex represents a turning point where the parabola changes direction.
In the standard form equation of a parabola, such as \( y = ax^2 \), the vertex is located at the origin, or point \((0, 0)\). This is because there are no \( x \)- or \( y \)-terms added or subtracted which would otherwise shift the vertex from its original position.
Finding the vertex is crucial in graphing a parabola as it helps to understand its structure and symmetry. For the parabola given by \( 4x^2 = -2y \), after rewriting as \( y = -2x^2 \), we see that the vertex remains at the origin \((0, 0)\) and serves as a pivotal guide in constructing the graph.

Equation of a Parabola

The equation of a parabola provides a formulaic representation of its shape and orientation. Specifically, we can express the equation in several forms, but one of the simplest is the standard form, \( y = ax^2 \), where the term \( a \) helps us determine the width and direction of the parabola.
In the given equation \( 4x^2 = -2y \), rewriting it as \( y = -2x^2 \) gives us the standard form. Here, \( a = -2 \), indicating that the parabola is vertically oriented and opens downward because \( a \) is negative.
Understanding the equation of a parabola allows you to easily plot key points by substituting \( x \)-values, like \( x = 1 \) or \( x = -1 \), and calculating corresponding \( y \)-values to place on the graph. This process forms a clearer visualization of how the parabola behaves in both expansive and detailed views.

Axis of Symmetry of a Parabola

The axis of symmetry of a parabola is a vital concept as it divides the parabola into two mirroring halves. It is a vertical line that runs through the vertex, ensuring that each point on one side of the axis aligns perfectly with a corresponding point on the opposite side.
In the case of a parabola given by \( y = ax^2 \), the axis of symmetry is always the line \( x = 0 \), as it passes through the vertex located at \((0, 0)\).
This vertical line is not only essential for symmetrical analysis but also serves a practical role in graph sketching. If you determine the axis of symmetry first, plotting points becomes more straightforward since you can reflect them easily about \( x = 0 \). Understanding this fundamental aspect can significantly enhance how effectively you can visualize and draw parabolas on a graph.