Problem 14 - Algebra and Trigonometry

Question

\(9-18\) m 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

Domain: \((-\infty, \infty)\), Range: \([4, \infty)\).

Step 1: Understand the Function

The given function is \( f(x) = x^2 + 4 \). This is a quadratic function of the form \( ax^2 + bx + c \) with \( a = 1 \), \( b = 0 \), and \( c = 4 \). Quadratic functions generally produce parabolic graphs that open upwards (since \( a > 0 \)).

Step 2: Plot the Function Using a Graphing Calculator

Input the function \( f(x) = x^2 + 4 \) into a graphing calculator or an online graphing tool. Observe the shape and position of the parabola on the graph. This parabola will be symmetric about the y-axis due to the absence of a linear term \( x \). It will have a vertex at \( (0, 4) \) and will open upwards.

Step 3: Determine the Domain

The domain of a function is the set of all possible input values (\( x \)-values). For \( f(x) = x^2 + 4 \), you can input any real number because squaring a real number always produces a valid result. Hence, the domain is all real numbers \( (-\infty, \infty) \).

Step 4: Determine the Range

The range of a function is the set of all possible output values (\( y \)-values). Since the lowest point of the parabola is the vertex at \( (0, 4) \) and it opens upwards, the minimum value of \( f(x) \) is \( 4 \). Therefore, the range is \( [4, \infty) \).

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that is a graphical representation of a quadratic function. In its standard form, a quadratic function is expressed as \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. The coefficient \( a \) determines the direction and width of the parabola.

• When \( a > 0 \), the parabola opens upwards, like in our example \( f(x) = x^2 + 4 \).
• When \( a < 0 \), the parabola opens downwards.

For the function \( f(x) = x^2 + 4 \), the graph is perfectly symmetrical about the y-axis. This symmetry occurs because there is no \( x \) term (i.e., \( b = 0 \)).
The vertex of the parabola, which is also the lowest point when the parabola opens upwards, is located at \( (0, 4) \). The vertex provides valuable information about the parabola's position and can help determine the function's range.

Domain and Range

Understanding the domain and range is crucial to interpreting any function. The domain refers to all possible input values (x-values) for which the function is defined. For quadratic functions like \( f(x) = x^2 + 4 \), the domain is all real numbers. This means you can substitute any real number into \( f(x) \) and get a valid result.

• The domain of \( f(x) = x^2 + 4 \) is \( (-\infty, \infty) \), indicating that there are no restrictions on \( x \).

The range, on the other hand, refers to all possible output values (y-values) produced by the function. Because the parabola opens upwards and the vertex is the minimum point at \( y = 4 \), the range starts at 4 and extends to infinity.

• Thus, the range of \( f(x) = x^2 + 4 \) is \( [4, \infty) \).

Recognizing the domain and range of a function allows you to understand its behavior and limitations better.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions, especially complex ones like quadratic functions. To graph \( f(x) = x^2 + 4 \) using a graphing calculator, simply enter the function into the calculator. The calculator will display a graph showing the characteristic U-shape of the parabola, which helps in understanding the features of the function more clearly.
Using a graphing calculator can assist in:

• Identifying the symmetry of the parabola about the y-axis due to the absence of a linear term in the function.
• Locating the vertex at \( (0, 4) \), a crucial step for determining the range.
• Observing the opening direction of the parabola, confirming that the graph opens upwards when \( a > 0 \).

Having a visual representation of the function allows you to easily identify key characteristics such as the vertex, axes of symmetry, domain, and range. Thus, a graphing calculator is an invaluable aid for students and professionals alike in studying functions.