Textbook Question
[Technology Exercise]
a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which
x/2 > 1 + 4/x
b. Confirm your findings in part (a) algebraically.
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0. Functions
Common Functions
2:30 minutesProblem 5.3.111a
Textbook Question
Zero net area Consider the function Ζ(π) = πΒ² β 4π .
(a) Graph Ζ on the interval π β₯ 0.
Verified step by step guidance1
Step 1: Start by analyzing the given function Ζ(π) = πΒ² - 4π. 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 Ζ'(π) = d/dπ [πΒ² - 4π] = 2π - 4. Set Ζ'(π) = 0 to solve for π, which gives the critical points.
Step 3: Determine the vertex of the parabola. The vertex occurs at π = -b/(2a) for a quadratic function in the form axΒ² + bx + c. Here, a = 1 and b = -4, so the vertex is at π = 2. Evaluate Ζ(2) to find the corresponding y-coordinate of the vertex.
Step 4: Identify the x-intercepts by solving Ζ(π) = 0. Factorize the quadratic equation πΒ² - 4π = 0 as π(π - 4) = 0, which gives the solutions π = 0 and π = 4. These are the points where the graph crosses the x-axis.
Step 5: Plot the graph of Ζ(π) on the interval π β₯ 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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 Ζ(π) = πΒ² - 4π, this means calculating y-values for various x-values, particularly within the specified interval x β₯ 0, and connecting these points to form a continuous curve.
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Finding Roots
Finding the roots of a function refers to determining the values of x for which the function equals zero. For Ζ(π) = πΒ² - 4π, this involves solving the equation πΒ² - 4π = 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.
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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 Ζ(π) = πΒ² - 4π 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.
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