Problem 30 - 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. $$ f(x)=-x^{2} $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\). Range: \((-\infty, 0]\).

Step 1: Identify the Given Function

The function provided is \( f(x) = -x^2 \). This is a quadratic function that opens downwards because the coefficient of \( x^2 \) is negative.

Step 2: Use a Graphing Calculator

Enter the function \( f(x) = -x^2 \) into your graphing calculator. Observe the shape of the graph. It should be a parabola opening downwards, with its vertex at the origin (0,0).

Step 3: Determine the Domain from the Graph

The domain of a function refers to all the possible input values (x-values) the function can have. For a quadratic function like \( -x^2 \), the graph extends infinitely in both the positive and negative x-directions. Thus, the domain is all real numbers, which can be expressed as \( (-\infty, \infty) \).

Step 4: Determine the Range from the Graph

The range of a function consists of all possible output values (y-values). Since the parabola opens downward and its highest point is at the vertex (0,0), the y-values go from negative infinity up to 0. Therefore, the range is \( (-\infty, 0] \).

Key Concepts

Function Graphing

Graphing a function can be a powerful way to visualize how it behaves. Let's take the example of the quadratic function \( f(x) = -x^2 \). This function is a type of parabola, which is a common form of graph seen in quadratic equations. When graphing \( f(x) = -x^2 \), you'll notice the graph forms a U-shape that opens downwards. This is because the coefficient of \( x^2 \) is negative, flipping the parabola upside down.

Using a graphing calculator, you can easily plot this function to better understand its shape. Here are some tips for using a graphing calculator for quadratic functions like this:

• Ensure the calculator is in the right mode (usually 'function' mode for graphing).
• Input the function correctly; remember the negative sign in front of \( x^2 \) is crucial here to get the right graph.
• Check the scale on the graph to make sure you're seeing the full parabola, especially the vertex.

Visually analyzing this graph, you'll see that it peaks at the vertex, which in this case is at the origin \((0, 0)\). The symmetry of the graph is also evident, with the y-axis serving as a line of symmetry.

Domain and Range

Understanding the domain and range of a function is essential for knowing where the function exists and how it behaves. For the function \( f(x) = -x^2 \), the domain and range can be determined by looking at the graph.

The **domain** of a quadratic function like \( f(x) = -x^2 \) is all real numbers. This is because there is no restriction on the x-values you can plug into the function. You can think of the domain as the set of all possible inputs, and for quadratic functions, the x-values extend infinitely in both the positive and negative directions. This is expressed mathematically as \((-\infty, \infty)\).

The **range** of \( f(x) = -x^2 \) is determined by the behavior of the y-values, or outputs. Since the parabola opens downwards, the highest value it reaches is at its vertex, which is 0. The y-values decrease as you move away from the vertex, going towards negative infinity. Therefore, the range encompasses all the y-values from negative infinity up to 0, expressed as \((-\infty, 0]\). This illustrates that while the x-values continue indefinitely, the y-values are capped at the vertex.

Parabolas

Parabolas are the graphical representation of quadratic functions, and they come in distinct shapes based on the sign of the coefficient of \( x^2 \). For example, the function \( f(x) = -x^2 \) results in a downward-opening parabola. But if the equation were \( f(x) = x^2 \), the parabola would open upwards.

Here are some key features of parabolas:

• **Vertex:** The highest or lowest point on the graph. For \( -x^2 \), it is the highest point, and for \( x^2 \), it is the lowest point.
• **Axis of symmetry:** A vertical line that divides the parabola into two mirror images. For \( f(x) = -x^2 \), the axis of symmetry is the y-axis or \( x = 0 \).
• **Direction:** A negative coefficient means the parabola opens downwards, while a positive coefficient means it opens upwards.

By understanding these characteristics, you can better interpret the graph of any quadratic function and predict its behavior. The nature of the parabola tells you a lot about how the function changes across different x-values, which is crucial for solving real-world problems and mathematical equations.