Problem 2 - Algebra A Combined Function

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. See Examples 1 through 4. $$ x=5 y^{2} $$

Step-by-Step Solution

Answer

The vertex is at (0, 0), and the parabola opens to the right.

Step 1: Identify the Standard Form

The given equation is \( x = 5y^2 \). Recognize that this equation represents a parabola. Parabolas can be represented in a standard form, which for parabolas opening to the left or right is \( x = a(y - k)^2 + h \), where \((h, k)\) is the vertex.

Step 2: Rewrite the Equation in Standard Form

To find the vertex, compare \( x = 5y^2 \) to the standard form \( x = a(y - k)^2 + h \). Here, \( a = 5 \), \( h = 0 \), and \( k = 0 \) since there are no additional terms to shift \( y \) or \( x \). Thus, the standard form is \( x = 5(y - 0)^2 + 0 \).

Step 3: Identify the Vertex

From the standard form \( x = 5(y - 0)^2 + 0 \), the vertex \((h, k)\) is \((0, 0)\). Thus, the vertex of the parabola is at the origin.

Step 4: Graph the Parabola

To graph the parabola, plot the vertex at \((0, 0)\). Since \( x = 5y^2 \), the parabola opens to the right because the coefficient of \( y^2 \) is positive. Plot additional points by selecting values for \( y \) and solving for \( x \) to ensure the shape of the parabola is accurate.

Key Concepts

Graphing Parabolas

Graphing a parabola is like turning a simple equation into a visual picture. To start, identify the vertex, which acts as the anchor point of the parabola. Once the vertex is plotted, remember that the direction in which the parabola opens depends on the equation's coefficients. For equations like \( x = ay^2 \), if \( a \) (the coefficient) is positive, the parabola opens to the right. If \( a \) is negative, it opens to the left. This equation differs from the vertical parabolas (i.e., \( y = ax^2 \)) where the parabolas open upwards and downward instead.When graphing, the vertex becomes your starting point. After plotting it, you can choose positive and negative values for \( y \) to find corresponding \( x \) values, helping you plot additional points. Connecting these points gives you the parabolic shape on the graph.

Standard Form of Parabolas

Understanding the standard form is crucial for easily identifying features of a parabola. Parabolas can have different orientations: some open up or down, others open to the left or right. The standard form for a horizontal parabola, which opens left or right, is \( x = a(y - k)^2 + h \).In this form:

• \( a \) is a coefficient that indicates the direction and width of the opening.
• \( (h, k) \) is the vertex, giving the parabola's precise position on the graph.

Horizontal parabolas are less common in basic algebra courses compared to vertical parabolas \( y = ax^2 + bx + c \), but they operate under similar principles. Identifying these elements helps rewrite equations into their most useful form for graphing or further analysis.

Vertex of a Parabola

The vertex of a parabola is a pivotal point on its graph. It represents the extreme point, often known as the minimum or maximum point, depending on the parabola's orientation.For parabolas with horizontal orientation, like \( x = ay^2 \), the vertex is represented as \( (h, k) \) within the standard form equation \( x = a(y - k)^2 + h \). You determine \( (h, k) \) by identifying any shifts in \( y \) and \( x \) from the standard position.In cases where a parabola hasn't been shifted, it might have a vertex at the origin, such as \( (0, 0) \). After finding the vertex, use it as a reference to plot the rest of the parabola efficiently, understanding that from this point, the graph will symmetrically curve in the specified direction.