Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ x=-2 y^{2} $$

Step-by-Step Solution

Answer

The graph is a leftward-opening parabola centered at (0, 0), with the origin as the intercept.

Step 1: Understand the Equation

The given equation is \( x = -2y^2 \). This is a quadratic equation in terms of \( y \) where \( x \) depends on \( y^2 \). This equation represents a parabola.

Step 2: Determine the Orientation

Since the equation is of the form \( x = Ay^2 \) where \( A = -2 \), this implies the parabola opens to the left. The negative sign indicates a leftward opening.

Step 3: Find the Vertex

The vertex of a parabola described by \( x = Ay^2 \) is at the origin \((0, 0)\) when there are no additional terms. Thus, the vertex for this equation is at the point \((0, 0)\).

Step 4: Identify the Intercepts

To find the \(x\)-intercept, set \( y = 0 \): \( x = -2 \times 0^2 = 0 \). Thus, the \(x\)-intercept is \( (0, 0) \). The \( y\)-intercepts occur when \( x = 0 \). Setting \( -2y^2 = 0 \) and solving for \( y \), we get \( y = 0 \). Thus, the \(y\)-intercept is also \((0, 0)\).

Step 5: Sketch the Parabola

Plot the vertex \((0, 0)\), and sketch the parabola which opens to the left. Since it is centered on the origin and opens leftwards, the graph is symmetric around the \(x = 0\) line but stretches horizontally along \(y\).

Step 6: Label the Intercepts on the Graph

The only intercept is the origin \((0, 0)\), which serves as both the \(x\)-intercept and \(y\)-intercept. Label this point on your graph accordingly.

Key Concepts

Understanding the x-intercept

An x-intercept is a point where a graph crosses the x-axis. This happens when the value of y is zero. In our quadratic equation, which is expressed as \( x = -2y^2 \), finding the x-intercept involves setting \( y \) to zero. When you substitute \( y = 0 \) into the equation:\[ x = -2 \times 0^2 = 0. \]
Thus, the x-intercept for this quadratic equation is at the point \((0, 0)\).
This tells us that the graph meets the x-axis at the origin.
Identifying the x-intercept is crucial for understanding the behavior of the parabola as it provides a reference point on the x-axis.

Identifying the y-intercept

A y-intercept is a point where the graph crosses the y-axis. This occurs when \( x \) is zero. For our quadratic equation \( x = -2y^2 \), we find the y-intercept by setting \( x \) to zero and solving for \( y \). This leads to the equation:
\( -2y^2 = 0 \). Simplifying this gives \( y = 0 \).
Hence, the y-intercept for this parabola is also \((0, 0)\).
Like the x-intercept, this indicates that the graph touches the y-axis at the origin.
Understanding the y-intercept is important as it helps to position the graph on the coordinate plane correctly.

Decoding the vertex of a parabola

The vertex of a parabola is a significant point as it represents the maximum or minimum value on the graph, depending on its orientation.
In our quadratic equation \( x = -2y^2 \), the vertex is at the origin \((0, 0)\).
This is because the equation is simplified to the form \( x = Ay^2 \) without any additional terms to move the vertex. The vertex is the pivot point of the parabola, around which the graph is symmetrical.
The orientation in our case (opening leftwards due to the negative sign) suggests that the vertex is a maximum point along the x-axis for this specific graph, verifying the central and symmetrical nature of the parabola around this point.

Exploring the quadratic equation

A quadratic equation is a second-degree polynomial typically of the form \( ax^2 + bx + c = 0 \) in standard expression terms.
However, the equation in this exercise, \( x = -2y^2 \), is represented differently where \( y \) is the squared variable. This structure forms a parabola, indicative of all quadratic equations.
These graphs come with distinctive features:

• They are symmetric;
• They have a U-shape (sometimes inverted);
• And depending on whether the leading coefficient (in this case, \(-2\)) is positive or negative, they may open upwards or downwards (in the y-variable version, this relates to the x-direction for the graph).

Understanding this formulation helps in identifying the basic structure and behavior of the graph, including points such as the vertex, axis of symmetry, and intercepts. In this scenario, recognizing the negatives in the coefficient reveals why the parabola opens to the left, showing how pivotal these equations are in predicting the shape and characteristics of graphs.