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

검증된 단계별 안내

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).

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.

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.

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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