Problem 26 - Algebra 2

Question

For Exercises \(24-27,\) use the equation \(x=3 y^{2}+4 y+1\) What are the coordinates of the vertex?

Step-by-Step Solution

Answer

The vertex of the parabola is at \(\left(\frac{-1}{3}, \frac{-2}{3}\right)\).

Step 1: Identify the Parabola's Format

The given equation is \(x = 3y^2 + 4y + 1\). This is a quadratic equation in standard form for a parabola that opens horizontally. Our task is to find the vertex, which in a quadratic function \(x = ay^2 + by + c\), is located at \(y = -\frac{b}{2a}\).

Step 2: Calculate the y-coordinate of the Vertex

Use the formula \(y = -\frac{b}{2a}\) where \(a = 3\) and \(b = 4\). Substituting these values gives: \[y = -\frac{4}{2 \times 3} = -\frac{4}{6} = -\frac{2}{3}.\]

Step 3: Calculate the x-coordinate of the Vertex

Substitute \(y = -\frac{2}{3}\) back into the original equation \(x = 3y^2 + 4y + 1\) to find the x-coordinate:\[x = 3\left(-\frac{2}{3}\right)^2 + 4\left(-\frac{2}{3}\right) + 1.\]Simplifying each term gives:\[x = 3 \times \frac{4}{9} - \frac{8}{3} + 1 = \frac{12}{9} - \frac{8}{3} + 1= \frac{4}{3} - \frac{8}{3} + 1.\]Combine the terms:\[x = -\frac{4}{3} + 1 = \frac{-4}{3} + \frac{3}{3} = \frac{-1}{3}.\]

Step 4: Write the Vertex Coordinates

Now that we have both coordinates, the vertex of the parabola is at \(\left(\frac{-1}{3}, \frac{-2}{3}\right)\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that is essential to understand in algebra. It involves a variable raised to the second power, making the equation look like this: \[ ax^2 + bx + c = 0 \]. Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). A quadratic equation graphs as a parabola, a U-shaped curve, which can open either upwards or downwards. Yet, when we have an equation like \( x = ay^2 + by + c \), it represents a parabola that opens horizontally instead. This shift in perspective is crucial when manipulating and solving quadratic equations in different forms.

Standard Form of a Parabola

The standard form of a parabola is a specific way to write quadratic equations to make them easier to analyze and graph. For parabolas that open horizontally, the standard form is written as \( x = ay^2 + by + c \). This form helps to easily identify the direction in which the parabola opens: horizontally with respect to the \( y \)-axis. Knowing the standard form allows us to find the vertex, axes of symmetry, and other significant properties of the parabola quickly. The coefficients \( a \), \( b \), and \( c \) directly influence the width, direction, and position of the parabola on the graph.

Finding Vertex

The vertex of a parabola is a critical point because it represents the peak or the lowest point, depending on the orientation. To find the vertex of a parabola given in the form \( x = ay^2 + by + c \), you can utilize the vertex formula: \( y = -\frac{b}{2a} \). This will give you the \( y \)-coordinate of the vertex. Once you have this \( y \)-value, you can substitute it back into the original quadratic equation to find the \( x \)-coordinate. For example, with the equation \( x = 3y^2 + 4y + 1 \), after substituting \( y = -\frac{2}{3} \), we find the \( x \)-coordinate to complete the vertex \( \left( \frac{-1}{3}, \frac{-2}{3} \right) \). Understanding these calculations allows you to pinpoint this vital feature of a parabola accurately.

Parabola Opens Horizontally

When dealing with parabolas, the orientation—whether they open horizontally or vertically—can change how you approach finding their properties. A parabola that opens horizontally, as seen in equations of the form \( x = ay^2 + by + c \), turns its vertex and line of symmetry around the \( y \)-axis. This means that the role of \( x \) and \( y \) can often feel reversed compared to usual quadratic equations. Horizontally opening parabolas are less common in standard mathematics curriculum compared to their vertically opening counterparts. However, they play an essential part in advanced topics such as hyperbolas or conic sections. Being familiar with this orientation helps in understanding broader mathematical contexts and applications.