Problem 7 - College Algebra

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

Step 1: Identify the Equation Type

The given equation is \( y = 4x^2 \). This is a quadratic equation because the highest degree of the variable \( x \) is 2. Quadratic equations in two variables describe parabolas.

Step 2: Recognize the Standard Form of a Parabola

The standard form for a parabola that opens upwards or downwards is \( y = ax^2 + bx + c \). The given equation already resembles this form, where \( a = 4 \), \( b = 0 \), and \( c = 0 \).

Step 3: Verify and Rewrite in Standard Form

Upon inspection, the equation \( y = 4x^2 \) is already in the standard form of a parabola equation since it matches \( y = ax^2 + bx + c \) with values \( b = 0 \) and \( c = 0 \). There are no additional terms to simplify or rearrange.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra that involve polynomials of degree 2. They take the general form of \( ax^2 + bx + c = 0 \), where:

• \( a \), \( b \), and \( c \) are constants.
• \( a eq 0 \), otherwise the equation becomes linear, not quadratic.
• \( x \) represents the variable.

One of the key features of quadratic equations is their ability to describe the shape of a parabola when plotted on a graph. The solutions to quadratic equations are typically found using methods like factoring, completing the square, or applying the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where the discriminant, \( b^2 - 4ac \), provides insight into the number and type of solutions the quadratic equation has. Quadratic equations can have two real solutions, one real solution, or two complex solutions depending on the discriminant's value.

Standard Form of a Parabola

The standard form of a parabola is essential to understanding how quadratic equations relate to parabolic shapes. This form is expressed as \( y = ax^2 + bx + c \), which showcases several important elements:

• \( a \) determines the opening direction and width of the parabola. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• \( b \) and \( c \) influence the position and the symmetry of the vertex along the x-axis and y-axis, respectively.

Changing these coefficients can move the parabola around the graph, stretch or compress it, and invert it. Recognizing this standard form enables students to quickly determine the general behavior and characteristics of a parabola, helping them graph it more easily.

Equation Identification

Identifying whether an equation represents a parabola involves looking for specific characteristics indicative of quadratic equations. The primary clue is the presence of the squared variable term, as seen in the general form \( y = ax^2 + bx + c \). To confirm that an equation such as \( y = 4x^2 \) is a parabola, consider these points:

• The highest degree of the variable is 2, confirming it's quadratic.
• The equation can be rewritten in the standard form with coefficients \( a = 4 \), \( b = 0 \), and \( c = 0 \).
• There are no additional operations altering the basic structure of \( ax^2 + bx + c \).

Understanding how to identify this form quickly is crucial for recognizing parabolas and working efficiently with quadratic equations in mathematical contexts.