Problem 32 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=x^{2}+4 $$

Step-by-Step Solution

Answer

The domain is \( (-\infty, \infty) \) and the range is \( [4, \infty) \).

Step 1: Graph the Function

Using a graphing calculator, plot the function \( f(x) = x^2 + 4 \). This will result in a parabola opening upwards, with its vertex at the point (0,4). The graph will be symmetric around the y-axis.

Step 2: Determine the Domain

The domain of a function corresponds to all the possible x-values the function can accept. Since \( f(x) = x^2 + 4 \) is a quadratic function, it can accept any real number as input. Therefore, the domain is all real numbers, which can be expressed in interval notation as \( (-\infty, \infty) \).

Step 3: Determine the Range

The range of a function corresponds to all the possible y-values a function can output. For \( f(x) = x^2 + 4 \), since the minimum value of \( x^2 \) is 0 and it increases from there, the minimum output value is \( 4 \). Therefore, the range is all real numbers greater than or equal to 4, written in interval notation as \( [4, \infty) \).

Key Concepts

Graphing Parabolas

Parabolas are a special shape in mathematics that we often see in quadratic functions. When you graph a quadratic function like \( f(x) = x^2 + 4 \), you get a curve called a parabola. In this case, because it is in the form \( f(x) = ax^2 + bx + c \), it will produce a U-shaped curve that opens upward because the coefficient of \( x^2 \), which is \( a \), is positive. This graph is symmetrical, meaning if you draw a vertical line through the middle of it, both sides will be mirror images.

Important Points about Graphing Parabolas:

• The vertex is a key point on the graph. For this particular function, the vertex is at (0, 4).
• Parabolas have an axis of symmetry, a vertical line that runs through the vertex. Here it is the y-axis, or \( x = 0 \).
• Each parabola opens either upwards or downwards based on the leading coefficient \( a \). If \( a > 0 \), it opens upwards, as with \( x^2 + 4 \).

Using a graphing calculator makes plotting these points more precise, allowing you to see the full shape of the parabola beyond just a few calculated points.

Domain and Range

When dealing with quadratic functions, understanding the domain and range is crucial. The domain of a function refers to all the possible x-values that will work in the function, while the range refers to all possible y-values that the function can output.

Domain:

• For quadratic functions like \( f(x) = x^2 + 4 \), the domain is all real numbers because you can place any real number into \( x \) and calculate \( f(x) \).
• This can be written in interval notation as \( (-\infty, \infty) \).

Range:

• The range here is slightly different. Since the parabola opens upward and has its lowest point at the vertex, \( y \) values will always be at least 4, the y-coordinate of the vertex.
• Thus, the range of \( f(x) = x^2 + 4 \) is \( [4, \infty) \), meaning all real numbers 4 or greater.

By analyzing the graph, students can visually confirm the domain and range, enhancing their understanding of these foundational concepts.

Vertex of a Parabola

The vertex of a parabola is a crucial point that tells us a lot about the graph. It can be seen as the highest or lowest point on the parabola, depending on its orientation. For a function like \( f(x) = x^2 + 4 \), the vertex is found directly from the function itself.

Understanding the Vertex:

• The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \). It's easy to see the transformation points \( (h, k) \) as the vertex.
• In \( f(x) = x^2 + 4 \), the vertex is \( (0, 4) \) because it can be rewritten as \( f(x) = 1(x-0)^2 + 4 \).
• The x-coordinate of the vertex provides the axis of symmetry, a line that splits the parabola equally, which in our example is \( x = 0 \).

Knowing the vertex aids in drawing the graph accurately and understanding the transformation of the parabola's shape. It provides a foundation point from which other behaviors of the parabola are derived, such as the direction in which it opens and its minimum or maximum value.