Problem 12 - 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. $$ f(x)=x^{2}, \quad-2 \leq x \leq 5 $$

Step-by-Step Solution

Answer

Domain: [-2, 5]. Range: [0, 25].

Step 1: Understand the Function

The given function is a quadratic function: \( f(x) = x^2 \), defined over the interval \(-2 \leq x \leq 5\). This means we only consider the section of the parabola within these boundaries.

Step 2: Graph the Function

Using a graphing calculator, plot the function \( f(x) = x^2 \) over the domain \([-2, 5]\). You should see a parabola opening upwards with its vertex at the origin (0,0).

Step 3: Determine the Domain

The domain of \( f(x) \) is given directly in the problem as the values for which \( x \) is defined. Therefore, the domain is \([-2, 5]\). This is simply all x-values between -2 and 5, inclusive.

Step 4: Determine the Range

To find the range, identify the minimum and maximum values of \( f(x) \) on the given interval. Since \( f(x) = x^2 \) is a parabola opening upwards, the minimum value occurs at the vertex and is 0 (when \( x = 0 \)). The maximum value occurs at \( x = 5 \), which gives \( f(5) = 25 \). Therefore, the range is \([0, 25]\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial that can be expressed in the form \( f(x) = ax^2 + bx + c \). In this expression, \( a, b, \) and \( c \) are constants, with \( a eq 0 \). Quadratic functions create a unique U-shaped graph known as a parabola. These functions are notable for their symmetry and simple shape. To understand why a quadratic function behaves the way it does, focus on the \( ax^2 \) term. This term determines the direction and width of the parabola:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Given our function \( f(x) = x^2 \), it is a standard quadratic where \( a = 1 \), and there are no linear or constant terms \((b = 0, c = 0)\). This simplifies our analysis, making it easy to visualize why the parabola opens upwards with its vertex at the origin (0,0). The quadratic function is key in solving practical problems involving areas, trajectories, and other phenomena in physics and engineering.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing mathematical functions. It helps plot functions to understand their behavior over a particular domain. Using a graphing calculator can simplify complex calculations and provide quick insights into how functions behave. For functions like \( f(x) = x^2 \), they allow students to see the U-shaped curve of the parabola almost instantly. Here's how a graphing calculator can assist in understanding the quadratic function:

• Input the quadratic function, and specify the domain, like \([-2, 5]\).
• Observe the shape of the parabola, seeing the symmetrical nature and vertex location.
• Identify key graphical features like the vertex (minimum point for an upward parabola) and intercepts.

In education, a graphing calculator can bridge the gap between formula understanding and visual data representation. It can cater to students who learn better visually, by providing a dynamic way to explore math concepts.

Parabola

A parabola is the graph that represents a quadratic function like \( f(x) = x^2 \). Its distinctive U-shape is due to the squared term in the function. It is symmetric around a line called the axis of symmetry. For the function \( f(x) = x^2 \), this axis is the y-axis itself.A parabola has several defining characteristics:

• Vertex: The lowest point (for \( a > 0 \)) or highest point (for \( a < 0 \)) on the parabola. For the function \( f(x) = x^2 \), the vertex is at (0, 0).
• Direction: Determined by the sign of \( a \). Our parabola opens upwards because \( a = 1 \).
• Focus and Directrix: These are advanced concepts related to the parabola's geometric definitions but not essential for basic understanding with graphing calculators.

Seeing a parabola, whether drawn by hand or on a graphing calculator, offers a way to appreciate the mathematical symmetry and explore other real-life applications like projectiles in physics.

Function Graph

A function graph is a visual representation of mathematical expressions. Graphs of functions tell us at a glance how variables are related and how they change. With a quadratic function like \( f(x) = x^2 \), plotting the function graph allows us to observe the intricate properties of the parabola across its domain.To construct the graph of \( f(x) = x^2 \) manually, follow these steps:

• Choose several values for \( x \) within the domain, say from -2 to 5.
• Calculate the corresponding \( f(x) \) values to establish coordinates (e.g., if \( x = 1 \), \( f(1) = 1^2 = 1 \)).
• Plot these points on a Cartesian plane and join them smoothly, respecting the direction and curvature.

When using a graphing calculator, this process is automated, providing an accurate and quick portrayal of the graph. Observing the function graph helps find the function's domain and range:

• The domain for \( f(x) = x^2 \) was pre-set as \([-2, 5]\) due to the problem's conditions.
• The range is derived from the smallest and largest y-values visible on the graph, here being \([0, 25]\) from the vertex to the endpoint \( x = 5 \) (where \( f(5) = 25 \)).

A function graph is more than just a visual tool; it is an integral part of understanding mathematical relationships deeply.