Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=3√x-2

Verified step by step guidance

1. Identify the given function: \(f(x) = 3\sqrt{x} - 2\). This is a square root function multiplied by 3 and then shifted down by 2 units.
2. Recall the basic graph of \(y = \sqrt{x}\), which starts at the origin \((0,0)\) and increases slowly to the right.
3. Apply the vertical stretch by 3: multiply the output of \(\sqrt{x}\) by 3, so the graph becomes steeper. The new function before shifting is \(y = 3\sqrt{x}\).
4. Apply the vertical shift down by 2 units: subtract 2 from the entire function, resulting in \(y = 3\sqrt{x} - 2\). This moves every point on the graph down by 2.
5. To graph, plot key points such as \(x=0\) (where \(f(0) = 3\sqrt{0} - 2\)), \(x=1\), and \(x=4\), calculate their corresponding \(y\) values, then sketch the curve starting at \((0, -2)\) and increasing to the right.

Key Concepts

Understanding the Cube Root Function

The cube root function, denoted as f(x) = ∛x, is the inverse of the cube function. It produces real outputs for all real inputs and has a characteristic S-shaped curve passing through the origin. Recognizing its shape helps in graphing transformations accurately.

Function Transformations

Transformations involve shifting, stretching, or reflecting the graph of a base function. In f(x) = ∛(x) - 2, subtracting 2 shifts the graph vertically downward by 2 units. Understanding these shifts is essential to correctly position the graph on the coordinate plane.

Plotting Key Points and Using Symmetry

Plotting key points such as where x = 0, 1, and -1 helps in sketching the graph accurately. The cube root function is symmetric about the origin, so using symmetry can simplify graphing. These points guide the shape and position of the transformed graph.