Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(3 x^{2}+y=14\)

Step-by-Step Solution

Answer

Yes, \( y = 14 - 3x^2 \) is a function of \( x \).

Step 1: Isolate y in the equation

To determine if the relation represents \( y \) as a function of \( x \), we need to express \( y \) in terms of \( x \). Start by isolating \( y \) on one side of the equation:\[ 3x^2 + y = 14 \]Subtract \( 3x^2 \) from both sides to get:\[ y = 14 - 3x^2 \]

Step 2: Understand the definition of a function

A relation between \( x \) and \( y \) is a function if, for every value of \( x \), there is exactly one corresponding value of \( y \). This means \( y \) should be uniquely determined by \( x \).

Step 3: Analyze the equation

The expression \( y = 14 - 3x^2 \) is a quadratic function with respect to \( x \). It defines \( y \) uniquely for any given \( x \), since there are no other operations (like taking square roots, etc.) that could introduce ambiguity about the value of \( y \) for each \( x \).

Step 4: Conclusion

Since \( y \) is uniquely defined for every \( x \), the relation \( 3x^2 + y = 14 \) does represent \( y \) as a function of \( x \).

Key Concepts

Quadratic Functions

Quadratic functions are a central element of algebra, forming the basis of many advanced mathematical concepts. At their core, a quadratic function is a polynomial of degree two, which means it has the highest exponent of 2. The general form is:
\[y = ax^2 + bx + c\]where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). If \( a \) were zero, the function would be linear. In quadratic functions, the \( x^2 \) term is what gives the parabola its characteristic shape.

In the function \( y = 14 - 3x^2 \), the \( x^2 \) term carries a negative coefficient (-3), causing the parabola to open downwards. This means as you move along the x-axis, the value of \( y \) will decrease, creating a maximum point - the peak of the parabola.

• This peak occurs at the vertex, calculated using the formula \( x = -\frac{b}{2a} \).
• In our example, because \( b = 0 \), the vertex lies on the y-axis, precisely at \( x=0 \).

This vertex point is vital because it tells us where the maximum or minimum value of the quadratic function occurs.

Algebraic Equations

An algebraic equation is a mathematical statement where two expressions are set equal to each other. Algebra deals with finding unknown values that satisfy this condition. Typically, equations involve variables like \( x \) or \( y \), constants like numbers, and mathematical operations such as addition, subtraction, multiplication, or division.

In our original problem, the equation \( 3x^2 + y = 14 \) involves solving for \( y \) in terms of \( x \). Through basic algebraic manipulation:

• Subtract \( 3x^2 \) from both sides to isolate \( y \).
• This step simplifies the equation to \( y = 14 - 3x^2 \).

This form is very useful because it directly shows how \( y \) changes according to different values of \( x \). Understanding how to rearrange and solve equations like this is a fundamental skill in algebra. It helps you uncover the behavior of variables and their interrelationships.

Relation and Mapping

In mathematics, a relation is any set of ordered pairs. Each pair consists of an input value from a set \( X \), known as the domain, and an output value in a set \( Y \), called the range. A relation becomes a function when each input \( x \) has exactly one output \( y \).

The problem asked whether \( 3x^2 + y = 14 \) described a function. Once we rearranged it as \( y = 14 - 3x^2 \), it becomes clear:

• For every \( x \), there's a unique \( y \).
• There's no ambiguity for the value of \( y \) given any \( x \).

Mapping describes this relationship: every point \( (x, y) \) adheres to the same rule, ensuring that the equation represents a function. This consistent pairing underpins much of algebraic and pre-calculus studies, emphasizing the predictive power of mathematical functions.