Problem 26 - Algebra 1: Concepts and Skills

Question

The function \(y=x^{2}\) has a U-shaped graph called a parabola. If the domain of this function is given as \(x \geq 0\) and \(x \leq 4,\) find the range.

Step-by-Step Solution

Answer

The range of the function y = \(x^{2}\) for the domain \(0 \leq x \leq 4\) is \(0 \leq y \leq 16\).

Step 1: Understand the Function

Consider the function \(y=x^{2}\), a quadratic function that describes a U-shaped curve or parabola. For this function, the lowest point (vertex) is at (0,0) when x=0.

Step 2: Define the Domain

The domain of a function is the complete set of possible values of the independent variable. In this exercise, the given domain is \(x \geq 0\) and \(x \leq 4\). That means x takes any value from 0 to 4, inclusive.

Step 3: Find the Range

Substitute the endpoints of the domain into the function to find the range. When substituting x=0 into the function, the result is \(0^{2}=0\). When x=4 are substituted, the resulting y-value is \(4^{2}=16\). So, the range of the function for this domain is \(0 \leq y \leq 16\).

Key Concepts

Understanding Parabolas

A parabola is a symmetrical open plane curve formed by plotting a quadratic function like \( y = x^2 \). It resembles a "U" shape, and its understanding is crucial when studying quadratic functions. Each parabola is defined by its unique vertex and where it opens. Parabolas can open upwards, like \( y = x^2 \), or downwards if negated, like \( y = -x^2 \).

Key features of a parabola include:

• The **vertex**, which is the point where the parabola changes direction.
• The **axis of symmetry** that vertically divides the parabola into two mirror images.
• The **focus** and **directrix** that help define its precise shape.

The simplest parabola, \( y = x^2 \), has its vertex at the origin \( (0, 0) \), and opens upwards. Understanding these basic characteristics helps in graphing and analyzing the properties of quadratic functions.

Exploring Domain and Range

The **domain** and **range** are foundational concepts in mathematics that describe the input and output of a function. For the quadratic function \( y = x^2 \), the domain is simply the set of all possible \( x \) values. The standard domain of \( y = x^2 \) is all real numbers because any real number squared remains a real number. However, in this exercise, we've been given a limited domain: \( 0 \leq x \leq 4 \).

The **range** corresponds to the possible \( y \) values that result when the \( x \) values from the domain are plugged into the function. By squaring the values at the endpoints of the domain, \( 0^2 = 0 \) and \( 4^2 = 16 \), we find the range is \( 0 \leq y \leq 16 \), as \( y \) takes all values from 0 to 16 inclusive.

Remember:

• **Domain**: The set of all possible independent variable (usually \( x \)) values.
• **Range**: The resulting set of dependent variable (usually \( y \)) values after applying the function on the domain.

Understanding domain and range figures prominently when evaluating the behavior and limitations of functions.

Vertex of a Parabola

The vertex of a parabola is a pivotal point and is often the starting reference point when discussing quadratic graphs. For the function \( y = x^2 \), the vertex is at the point \( (0, 0) \). This is the lowest point on the graph of a parabola that opens upwards, and the highest point for one that opens downwards.

In general, the vertex of a parabola described by a quadratic function \( y = ax^2 + bx + c \) can be found using the formula for the \( x \)-coordinate of the vertex: \( x = -\frac{b}{2a} \).

Once the \( x \)-coordinate is known, substitute it back into the equation to find the corresponding \( y \)-coordinate. Thus, the vertex provides a key piece of information required to sketch the parabola or optimize quadratic expressions.

Points to remember:

• The **vertex** is located at the minimum or maximum point of the parabola, depending on its orientation.
• Alterations in the quadratic function can shift the vertex, changing its position on the graph.

The vertex forms the "heart" of the parabola, dictating its structure and symmetry.