Problem 37 - Elementary and Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) See Example 3. $$ y=4 x^{2} $$

Step-by-Step Solution

Answer

Yes, \( y = 4x^2 \) defines y to be a function of x.

Step 1: Understand the Criteria for a Function

Recall the definition of a function. A relation between two variables, typically x and y, is called a function if each input x corresponds to exactly one output y. In graphical terms, this means that any vertical line drawn through the graph of the equation will intersect the graph at most once.

Step 2: Analyze the Given Equation

Consider the equation \( y = 4x^2 \). This equation fits the form of a quadratic function, which is a type of polynomial function where each x value maps to exactly one y value. Upon examination, it can be understood that for each x value, once the computations are done, there is only one value possible for y.

Step 3: Visualize with a Graph

Visualize or sketch the graph of \( y = 4x^2 \). This is a parabola opening upwards with its vertex at the origin. Applying the vertical line test (drawing vertical lines across the graph), you will observe that each vertical line intersects the graph at most once, confirming that each x value has only one corresponding y value.

Step 4: Conclusion Based on Analysis

Based on the analysis and graphing of the equation \( y = 4x^2 \), it is confirmed that it satisfies the definition of a function.

Key Concepts

Vertical Line Test

The vertical line test is a useful tool to determine whether a relationship between two variables is a function. Imagine drawing countless vertical lines across a graph. If any of these lines intersect the graph more than once, the relation is not a function. This is because there would be a single input (x-value) pairing with multiple outputs (y-values), which violates the definition of a function.

• Helps check if a graph is a function
• Only single intersections are allowed
• Multiple intersections indicate the relation is not a function

Applying this test to the equation \( y = 4x^2 \) shows each vertical line crosses the graph only once, ensuring that the equation is indeed a function.

Quadratic Function

Quadratic functions are an essential type of mathematical function that form a parabola when graphed. A quadratic function is generally given by this expression: \( y = ax^2 + bx + c \), where \( a eq 0 \). The graph of these functions is a U-shaped curve called a parabola, which can open upwards or downwards. The position of the parabola and the direction in which it opens are controlled by the coefficients \( a, b, \) and \( c \).

• Recognizable by their characteristic parabolic shape
• Can open upwards or downwards depending on the sign of \( a \)
• The vertex is the peak or the lowest point of the parabola

For example, in the quadratic function \( y = 4x^2 \), the parabola opens upwards, and its vertex is at the origin (0,0). Each value of x gives a unique y, confirming that it's a function.

Polynomial Function

Polynomial functions are fundamental expressions in mathematics, composed of terms with non-negative integer exponents. They can represent a wide variety of curves depending on the highest power of the variable, which is known as the degree of the polynomial. A polynomial is expressed as follows: \( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \). Each "degree" has its characteristics:

• Degree 1: Linear functions, straight lines
• Degree 2: Quadratic functions, parabolas
• Higher degrees: More complex curves

Polynomial functions are crucial for modeling various phenomena because they can approximate behaviors of real-world activities. Quadratic functions like \( y = 4x^2 \) are a specific type of polynomial function, characterized by having a single term with \( x^2 \), making them the simplest form of polynomial beyond linear equations.