Problem 2 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The vertex of the parabola is at (0, 0), and it opens downwards.

Step 1: Convert to Standard Form

The given equation is \(x^2 = -y\). We can rewrite it in the form \(y = -x^2\), which is the standard form for a parabolic equation.

Step 2: Identify Vertex and Orientation

From the equation \(y = -x^2\), we identify that the vertex of the parabola is at \((0, 0)\) because there are no transformations applied to \(x\) or \(y\) other than the coefficient. The negative sign indicates that the parabola opens downwards.

Step 3: Find Additional Points

Choose values for \(x\) to find corresponding \(y\) values. For example, substituting \(x = 1\) gives \(y = -(1)^2 = -1\), and \(x = -1\) gives \(y = -(-1)^2 = -1\). Calculate additional points: if \(x = 2\), then \(y = -(2)^2 = -4\) and similarly, if \(x = -2\), then \(y = -4\).

Step 4: Plot Points and Sketch the Graph

Plot the vertex \((0, 0)\) and the additional points \((1, -1)\), \((-1, -1)\), \((2, -4)\), and \((-2, -4)\) on a graph. Connect these points with a smooth curve to form a 'U' shaped graph opening downward. This is the graph of the parabola \(y = -x^2\).

Key Concepts

Vertex of a Parabola

In a parabola described by a quadratic equation in standard form, the vertex is the turning point of the curve. It is a pivotal feature that determines the parabola's maximum or minimum value, depending on its orientation. For the equation \(y = -x^2\), the vertex is particularly straightforward to identify because there are no other terms accompanying \(x\) or constant values that might shift it. This gives us a vertex at the origin, \((0, 0)\).
The vertex signifies the highest or lowest point on the parabola. Since the equation includes a negative coefficient before \(x^2\), this parabola opens downwards, making the vertex at \((0, 0)\) its highest point. Understanding the vertex's location, without translations, is crucial for graphing parabolas accurately.

Graph Sketching of Parabolas

Graph sketching involves laying out a visual representation of the quadratic equation by identifying key points and their connections. For \(y = -x^2\), the graph sketching process entails the following steps:

• Start with the vertex at \((0, 0)\) since it is the peak of the parabola.
• Recognize that the parabola opens downwards because of the negative coefficient.
• Select a few values of \(x\) to calculate corresponding \(y\) values, like \(x = 1\) and \(x = 2\), to get the points \((1, -1)\) and \((2, -4)\).
• Include similar calculations for negative \(x\) values like \(x = -1, -2\), which yield \((-1, -1)\) and \((-2, -4)\).

Plot these points on a coordinate plane. Draw a smooth curve through these points to create the classic 'U' shape of a parabola. Remember, the downwards orientation ensures it opens towards negative \(y\) values.

Understanding Quadratic Equations

Quadratic equations are polynomial expressions of degree 2, generally represented in the form \(ax^2 + bx + c = 0\). However, for graphing purposes, we convert them to the standard form \(y = ax^2 + bx + c\), as seen in \(y = -x^2\).
This form makes it easier to identify essential characteristics like:

• **The coefficient \(a\):** This value affects the direction and width of the parabola. A negative \(a\) in \(y = -x^2\) means the parabola opens downwards, showing how the graph behaves.
• **Verifying Symmetry:** Parabolas are symmetric around their vertex line, which lies on the \(y\)-axis for \(y = -x^2\).
• **Solving for \(x\):** Set up \(y\) values as zero to find \(x\)'s role in the roots of the equation, though our focus here is mainly on the graph's dimensional attributes.

By understanding these facets, drawing a parabolic graph becomes less complex, and properties like maximum or minimum values become clearer.