Problem 9 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Yes, \(y = x^2\) represents \(y\) as a function of \(x\).

Step 1: Recall the Definition of a Function

A relation is a function if each input value, or "x-value," is paired with exactly one output value, or "y-value." This means that for every value of \(x\), there should be only one corresponding \(y\).

Step 2: Analyze the Equation

Look at the equation \(y = x^2\). This is a quadratic equation, which is typically a parabolic curve when graphed. For every \(x\) that you plug into this equation, you can calculate \(y\) by squaring the \(x\) value.

Step 3: Test with Example Values

Choose a few example values for \(x\). For instance, if \(x = 2\), then \(y = 2^2 = 4\). Similarly, if \(x = -3\), then \(y = (-3)^2 = 9\). In both cases, each \(x\) value results in exactly one \(y\) value.

Step 4: Apply the Vertical Line Test

Although already sufficient, you can also use the vertical line test on the graph of \(y = x^2\). If you draw vertical lines across the graph, each line should intersect the graph in exactly one point for it to be a function. This is true for the graph of \(y = x^2\).

Key Concepts

Definition of a Function

A function is a fundamental concept in mathematics, particularly in algebra. Simply put, a function is a special type of relation. It connects each input value from a set of possible inputs, often called the domain, to a single output value in a set known as the range. The key characteristic of a function is that each input is paired with exactly one output.

When talking about functions, we often use "x-values" to refer to inputs and "y-values" for outputs. To determine if a relation is a function, we check whether each input has one and only one output.

• If an input ( x ight) can produce two different results for y ight), then it's not a function.
• If every x ight) has a unique y ight), then we have a function.

This is what makes functions so reliable when solving mathematical problems. They follow a strict rule where no input is left with uncertainty as to its output.

Quadratic Equation

A quadratic equation is a type of polynomial equation characterized by the highest power of the variable being 2. It has the general form: \[ax^2 + bx + c = 0\]where a ight), b ight), and c ight) are constants, and a eq 0 ight) ensures it's truly quadratic.

Quadratic equations create parabolas when graphed. These are U-shaped curves that open either upwards or downwards depending on the sign of a ight). If a > 0 ight), the parabola opens upwards. Conversely, if a < 0 ight), it opens downwards.
A specific example is the equation y = x^2 ight), which is a simple quadratic equation. Here, every x ight) value plugged into this equation will square it to give a single, unique y ight) value. This consistency is why quadratic equations so often represent functions.

Vertical Line Test

The vertical line test is a visual method people often use to determine if a graph represents a function. It's straightforward yet very effective in confirming function status.

To use the vertical line test:

• Draw vertical lines across the graph.
• If any vertical line touches the graph at more than one point, the graph does not represent a function.
• If each vertical line touches the graph at most once, the graph represents a function.

This test is particularly useful when analyzing graphs because it provides a quick confirmation of whether each input ( x ight) has one corresponding output ( y ight). For example, the graph of the function y = x^2 ight) passes this test cleanly, as every vertical line crosses the parabola at only one point.