Question

Graph each function. $$g(x)=\frac{1}{2} x^{3}+2$$

   Graph each function.
$$g(x)=\frac{1}{2} x^{3}+2$$

Precalculus

Margaret L. Lial,… 6th Edition

Chapter 2, Problem 82

Instant Answer

Solved by Expert Nick Johnson

05/26/2021

Step 1

This is a cubic function, which means it will have the general shape of a cubic curve, but it will be vertically compressed by a factor of $\frac{1}{2}$ compared to the standard cubic function $f(x) = x^3$. Show more…

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Graph each function. $$g(x)=\frac{1}{2} x^{3}+2$$

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Key Concepts

Vertical Transformation

Vertical transformations modify the output values of a function by stretching or compressing it. Multiplying the function by a constant affects the vertical scale; for example, a coefficient less than one compresses the graph vertically, altering its steepness while retaining its general shape.

Translation

Translation refers to shifting the graph of a function horizontally or vertically without changing its shape. Adding a constant to the function results in a vertical shift, moving the entire graph upward or downward, which affects features like the vertical intercept and overall positioning on the coordinate plane.

Graphing Functions

Graphing functions is the process of representing a function's output for various inputs on a coordinate system. It involves plotting points derived from the function's equation and understanding the behavior and notable features of the overall function such as intercepts, asymptotic behavior, and end behavior.

Cubic Functions

Cubic functions are polynomial functions of degree three, which typically exhibit an 'S'-shaped curve. They have characteristics such as an inflection point where the curve changes concavity, and the general shape is influenced by the leading coefficient, which affects the steepness and direction of the graph.

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