Question

Sketch the graph of the equation. $$ y=-x^{3}+1 $$

   Sketch the graph of the equation.
$$
y=-x^{3}+1
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 1, Problem 50

Instant Answer

Solved by Expert Nick Johnson

03/02/2022

Step 1

The graph of $y=x^{3}$ is a curve that increases for negative x values, passes through the origin (0,0), and continues to increase for positive x values. Show more…

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Sketch the graph of the equation. $$ y=-x^{3}+1 $$

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Key Concepts

Cubic Functions

Cubic functions are polynomial functions of degree three that typically exhibit an S-shaped curve with one point of inflection. Their behavior is controlled by the leading coefficient, which determines the end behavior of the graph, meaning how the graph behaves as x becomes very large or very small.

Vertical Reflection

Vertical reflection occurs when the sign of the function's output is reversed. In the context of cubic functions, multiplying by -1 flips the graph over the x-axis, so that the portions of the graph that originally went upward now go downward, and vice versa.

Vertical Translation

A vertical translation moves the graph up or down without changing its shape. Adding a constant to a function shifts every point vertically, so in this case, the graph is lifted upward by one unit, affecting the position of intercepts and other key features.

Graph Sketching Techniques

Graph sketching involves identifying key features such as intercepts, turning points, and end behavior. For cubic functions, understanding standard shapes and how transformations like reflections and translations affect these shapes is essential to accurately represent the graph.

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