Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .
(a) Graph ƒ on the interval 𝓍 ≥ 0.

Verified step by step guidance

1. Step 1: Start by analyzing the given function ƒ(𝓍) = 𝓍² - 4𝓍. Identify its key features, such as the degree of the polynomial (quadratic) and the leading coefficient (positive, indicating the parabola opens upwards).
2. Step 2: Find the critical points of the function by taking its derivative. Compute ƒ'(𝓍) = d/d𝓍 [𝓍² - 4𝓍] = 2𝓍 - 4. Set ƒ'(𝓍) = 0 to solve for 𝓍, which gives the critical points.
3. Step 3: Determine the vertex of the parabola. The vertex occurs at 𝓍 = -b/(2a) for a quadratic function in the form ax² + bx + c. Here, a = 1 and b = -4, so the vertex is at 𝓍 = 2. Evaluate ƒ(2) to find the corresponding y-coordinate of the vertex.
4. Step 4: Identify the x-intercepts by solving ƒ(𝓍) = 0. Factorize the quadratic equation 𝓍² - 4𝓍 = 0 as 𝓍(𝓍 - 4) = 0, which gives the solutions 𝓍 = 0 and 𝓍 = 4. These are the points where the graph crosses the x-axis.
5. Step 5: Plot the graph of ƒ(𝓍) on the interval 𝓍 ≥ 0. Mark the vertex, x-intercepts, and other key points. Sketch the parabola, ensuring it opens upwards and passes through the identified points.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values) of a function. For the function ƒ(𝓍) = 𝓍² - 4𝓍, this means calculating y-values for various x-values, particularly within the specified interval x ≥ 0, and connecting these points to form a continuous curve.

Finding Roots

Finding the roots of a function refers to determining the values of x for which the function equals zero. For ƒ(𝓍) = 𝓍² - 4𝓍, this involves solving the equation 𝓍² - 4𝓍 = 0, which can be factored to find the x-intercepts. These roots are critical for understanding where the graph intersects the x-axis and can indicate changes in the function's behavior.

Understanding Area Under the Curve

The area under the curve of a function on a given interval can provide insights into the function's behavior, such as net area, which accounts for regions above and below the x-axis. In this case, analyzing the graph of ƒ(𝓍) = 𝓍² - 4𝓍 will help determine if the net area is zero, which occurs when the positive and negative areas cancel each other out within the specified interval.