Problem 33 - Algebra 2

Question

Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function and state whether it is discrete or continuous. $$ y=x^{2} $$

Step-by-Step Solution

Answer

The domain is \((-\infty, \infty)\), range is \([0, \infty)\). It is a continuous function.

Step 1: Graph the Equation

The equation given is \(y = x^2\). This is a parabola that opens upwards. To graph it, plot several points: for example, \((x, y) = (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)\). Connect these points with a smooth curve to complete the graph.

Step 2: Determine the Domain

The domain of \(y = x^2\) is the set of all possible x-values. Since this is a basic quadratic function and there are no restrictions on x, the domain is all real numbers. In interval notation, this is \((-\infty, \infty)\).

Step 3: Determine the Range

The range is the set of all possible y-values. Since the parabola opens upwards and the vertex is at \(y = 0\), the smallest value y can take is 0. Thus, the range is \([0, \infty)\).

Step 4: Determine if it is a Function

For an equation to be a function, each x-value should map to exactly one y-value. The equation \(y = x^2\) satisfies this condition because for any x, there is only one y-value. Therefore, it is a function.

Step 5: Determine if it is Discrete or Continuous

A relation or set of points is discrete if it consists of distinct, separate points. It is continuous if you can draw it without lifting your pencil. Since \(y = x^2\) is a parabola, it is a smooth curve and therefore continuous.

Key Concepts

Domain and Range

Understanding the domain and range is crucial when working with quadratic functions like the equation \(y = x^2\). The domain refers to the complete set of possible input values (x-values), which for this function extends across all real numbers. This means you can choose any x-value, either positive, negative, or zero, and it will produce a corresponding y-value.
This can be represented in interval notation as \((-fty, fty)\).

• If there are no restrictions on the variable x, the domain will often be all real numbers.
• In quadratic functions, this is typical behavior unless specified otherwise by the problem.

The range, on the other hand, involves the possible output values (y-values). Since the curve \(y = x^2\) opens upwards and achieves its minimum at \(y = 0\), the range of the function is from 0 to infinity. Thus, the range is noted as \([0, fty)\), signifying that while y can reach any value above zero, it can never be negative.

Graphing Parabolas

Graphing a quadratic function like \(y = x^2\) results in a special U-shaped graph known as a parabola. For \(y = x^2\), the parabola opens upwards due to the positive coefficient of the \(x^2\) term. To sketch this, it's helpful to plot some key points.
The simplest approach is to calculate several values like:

• When \(x = -2\), \(y = 4\)
• When \(x = -1\), \(y = 1\)
• When \(x = 0\), \(y = 0\)
• When \(x = 1\), \(y = 1\)
• When \(x = 2\), \(y = 4\)

Plot these points on a coordinate grid and smooth them out into a smooth curve to see the parabola in full view. Notice the vertex is at the origin point \((0,0)\), and the curve is symmetric about the y-axis. The symmetry is a characteristic trait of parabolas, indicating that if you fold the graph along the y-axis, both halves will align perfectly.

Discrete vs Continuous Functions

In mathematics, distinguishing between discrete and continuous functions can help understand the behavior of their graphs. Discrete relations are composed of distinct, separate points. You can imagine them as a series of individual dots that aren't connected. Contrast this with continuous functions, which are represented by unbroken lines or curves.

• For discrete graphs, think of plotting data points from a survey: they lie separately.
• In continuous graphs, such as the parabola \(y = x^2\), the values of x and y fill every point on the curve.

The equation \(y = x^2\) clearly forms a continuous function because you draw the curve without lifting your pencil off the paper.
This seamless trait allows for an infinite number of points and means that between any two points on the graph, there exists another point. It’s essential to recognize the continuity in quadratic functions, as it influences other characteristics, such as domain and range.