Problem 13 - College Algebra

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). $$ y=-2 x^{2}+40 x $$

Step-by-Step Solution

Answer

Yes, the relation represents \(y\) as a function of \(x\).

Step 1: Identify the Form of the Equation

The given equation is in the form of a quadratic equation, which can be identified as a typical polynomial expression: \[ y = -2x^2 + 40x \] This equation is a polynomial of degree 2.

Step 2: Understand What a Function Is

A relation is a function if each input value corresponds to exactly one output value. For an equation to represent \(y\) as a function of \(x\), for every \(x\) value, there must be precisely one \(y\) value.

Step 3: Check the Parabolic Form

In a quadratic equation like \(y = -2x^2 + 40x\), the graph is a parabola opening either up or down since the highest power of \(x\) is 2. A parabola passes the vertical line test, meaning for every \(x\) there is one unique \(y\), unless \(x\) values repeat, which does not occur in equations of this form.

Step 4: Apply the Vertical Line Test

The vertical line test states that if a vertical line intersects the graph of the equation at no more than one point at any given \(x\) value, the relation is a function. Since any vertical line drawn through the parabola formed by \(y = -2x^2 + 40x\) intersects it at most once, this confirms the relation as a function.

Key Concepts

Parabola

In a graphing context, a _parabola_ is the U-shaped curve that forms when we plot a quadratic equation, such as the one given in the exercise, which is \[ y = -2x^2 + 40x \] This particular parabola will open downwards. Why? Because the coefficient of the squared term, \( -2x^2 \), is negative, which indicates the parabola 'sags' in the middle.
Parabolas are a crucial part of quadratic functions, always arching either up or down, which reflects the nature of these functions.
A few interesting points about parabolas:

• **Vertex**: The highest or lowest point on the parabola, depending on its orientation.
• **Axis of Symmetry**: A vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• **Opening Direction**: Controlled by the sign of the coefficient of the squared term (positive for upward, negative for downward).

Understanding the shape and orientation of a parabola is crucial in graphing quadratic equations. It also helps in performing the vertical line test, which assesses whether the relation is a function.

Polynomial Equation

A _polynomial equation_ is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. The given equation \[ y = -2x^2 + 40x \] is a polynomial of degree 2, which is why it is often called a quadratic equation.
The polynomial equation shows how the variables and coefficients work together to form specific types of curves when graphed. Here are some key points about polynomial equations:

• **Degree of the Polynomial**: Defines the highest power of the variable in the equation. A degree of 2 means the graph will be a parabola.
• **Coefficients**: Numbers multiplied by the variables, which can deeply affect the shape and position of the graph on a coordinate plane.
• **Roots or Zeros**: Values of \(x\) that make the polynomial equal to zero, potentially representing the intersection with the x-axis.

Polynomial equations tell us a lot about the function's shape and behavior, playing a crucial role in determining the relationship of \(x\) and \(y\). They are foundational in further algebraic studies and calculus.

Vertical Line Test

The _vertical line test_ is a straightforward visual method to determine if a curve on a graph represents a function of \(x\). Here’s how it works: if any vertical line passes through the graph more than once at any place, the relation is not a function. For the equation \[ y = -2x^2 + 40x \], the graph is a parabola.Here’s how this test proves the relation is a function:

• **Intersects Once:** Since the parabola opens downward and does not loop or bend back on itself, a vertical line will only touch it once at any given \(x\).
• **Unique Output:** Each \(x\) has a single corresponding \(y\) value, which is the condition for a relation to be a function.
• **Simplicity:** This test can be applied to any graph quickly and acts as an immediate check for functionality.

Overall, the vertical line test is a powerful tool for students learning about functions, ensuring that each input \(x\) has one and only one output \(y\). It is simple yet effective for confirming a relation's validity as a function.