Graphs and Graphing


Graph the curves in Exercises 33–42.


y = 𝓍³ (8―𝓍 )

Verified step by step guidance

1. Identify the function to be graphed: \( y = x^3 (8 - x) \). This is a polynomial function, which can be expanded to \( y = 8x^3 - x^4 \).
2. Determine the critical points by finding the derivative of the function and setting it equal to zero. The derivative is \( y' = \frac{d}{dx}(8x^3 - x^4) = 24x^2 - 4x^3 \). Set \( 24x^2 - 4x^3 = 0 \) and solve for \( x \).
3. Factor the derivative: \( 4x^2(6 - x) = 0 \). This gives the critical points \( x = 0 \) and \( x = 6 \).
4. Evaluate the function at the critical points and endpoints to determine the behavior of the graph. Calculate \( y(0) \), \( y(6) \), and consider the behavior as \( x \to \pm \infty \).
5. Analyze the intervals determined by the critical points to understand where the function is increasing or decreasing. Use the first derivative test on intervals \((-\infty, 0)\), \((0, 6)\), and \((6, \infty)\) to determine the nature of each critical point (local maxima, minima, or saddle point).

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The given function y = x³(8−x) is a polynomial of degree 4, which can be expanded to y = 8x³ − x⁴. Understanding the behavior of polynomial functions, such as end behavior and turning points, is crucial for graphing them.

Factoring and Roots

Factoring involves expressing a polynomial as a product of its factors, which can help identify the roots or zeros of the function. For y = x³(8−x), the roots are x = 0 and x = 8, where the graph intersects the x-axis. These roots are essential for sketching the graph, as they indicate where the function changes sign and crosses the axis.

Graphing Techniques

Graphing techniques involve plotting key points, such as intercepts and turning points, and understanding the shape and symmetry of the graph. For y = x³(8−x), one should consider the roots, the behavior as x approaches infinity, and any symmetry. The function is symmetric about the y-axis, and its end behavior is determined by the highest degree term, −x⁴, indicating the graph falls to negative infinity as x approaches positive or negative infinity.