Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y=4 x^{2}\)

Step-by-Step Solution

Answer

Yes, \(y = 4x^2\) is a parabola and is already in standard form.

Step 1: Identify the Equation Type

The given equation is \(y = 4x^2\). This is a quadratic equation in the form \(y = ax^2\), which suggests it may represent a parabola.

Step 2: Confirm the Equation is a Parabola

A parabola in general is described by a quadratic equation. Since the given equation \(y = 4x^2\) has the highest degree as 2, it confirms that the equation represents a parabola.

Step 3: Rewrite the Equation in Standard Form

The standard form of a parabola that opens upwards or downwards is \(y = ax^2 + bx + c\). The given equation \(y = 4x^2\) already fits this form with \(b = 0\) and \(c = 0\), so it is already in standard form.

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic equation. Parabolas are found in various applications, such as physics and engineering, because of their unique properties. In mathematics, each parabola is associated with a quadratic equation of the form \( y = ax^2 + bx + c \). This mathematical graph opens either upwards or downwards and is centered around a specific point known as the vertex.

• **Vertex**: The highest or lowest point of the parabola, depending on its direction of opening.
• **Axis of Symmetry**: A vertical line that divides the parabola into two mirror-image halves, passing through the vertex.
• **Focus and Directrix**: Geometric properties that further define the shape and position of a parabola on a graph. These are more advanced concepts that get explored as your knowledge deepens in algebra.

Understanding these parts can help in visualizing the equation's graph and analyzing its behavior.

Standard Form

The standard form of a quadratic equation, which describes a parabola, is \( y = ax^2 + bx + c \). This structure is vital as it provides an immediate insight into the parabola's basic character and orientation. Here is what each component represents:

• \(a\): Dictates the parabolic curve's 'steepness' and determines its direction of opening (upward if \(a > 0\) and downward if \(a < 0\)).
• \(b\): Affects the symmetry and position of the axis but is \(0\) in our example, indicating the parabola is symmetric around the y-axis.
• \(c\): The y-intercept, where the parabola crosses the y-axis.

In the given equation \(y = 4x^2\), the parameters are clear: \(a = 4\), and both \(b = 0\) and \(c = 0\). This allows us to determine that the parabola is opening upwards.

Degree of a Polynomial

The degree of a polynomial is crucial in identifying the shape and nature of the graph it represents. In simpler terms, it is the highest power of the variable present in the polynomial. For quadratic equations like \(y = 4x^2\), the degree is 2, which classifies it as a second-degree polynomial.

• **Degree 1**: Represents linear equations, resulting in straight lines.
• **Degree 2**: Represents quadratic equations, forming parabolas.
• **Higher Degrees**: Lead to more complex curves, often with multiple turning points.

Recognizing the degree helps in predicting the kind of graph and its potential complexity. Here, knowing that the degree is 2 immediately suggests the graph will be a parabola.

Upward and Downward Opening Parabolas

The direction in which a parabola opens is influenced by the sign of the coefficient \(a\) in the quadratic equation \( y = ax^2 + bx + c \).

• **Upward Opening**: Occurs if \(a > 0\); the arms of the parabola extend upwards creating a 'smile' shape.
• **Downward Opening**: Happens when \(a < 0\); the parabola curves downwards forming a 'frown' shape.

In our exercise, the equation \(y = 4x^2\) has \(a = 4\), which is greater than zero. This means the parabola opens upwards. Observing the sign of \(a\) allows a quick determination of the parabola's orientation, which is essential when sketching or analyzing graphs. Understanding these opening directions helps in visualizing and interpreting real-world data situations that parabolas often model.