Graph each function. See Examples 6–8. _ ƒ(x) = 2√x + 1

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Identify the function to be graphed: \(f(x) = 2\sqrt{x} + 1\). This is a transformation of the basic square root function \(y = \sqrt{x}\).

Recall the basic shape of \(y = \sqrt{x}\), which starts at the origin \((0,0)\) and increases slowly to the right, only defined for \(x \geq 0\).

Apply the vertical stretch by a factor of 2 to the function, changing it to \(y = 2\sqrt{x}\). This means every output value of \(\sqrt{x}\) is multiplied by 2, making the graph steeper.

Apply the vertical shift upward by 1 unit, resulting in \(f(x) = 2\sqrt{x} + 1\). This moves the entire graph of \(2\sqrt{x}\) up by 1 on the y-axis.

Plot key points by choosing values of \(x\) (such as 0, 1, 4, 9), calculate \(f(x)\) for each, and then sketch the curve starting at \((0,1)\) and increasing to the right, reflecting the transformations.

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Understanding the Square Root Function

The square root function, √x, outputs the non-negative value whose square is x. It is defined only for x ≥ 0 in the real number system. Graphing this function involves plotting points where y = √x, resulting in a curve starting at the origin and increasing slowly.

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Imaginary Roots with the Square Root Property

Function Transformations: Vertical Stretch and Shift

Multiplying a function by a constant greater than 1, like 2√x, vertically stretches the graph, making it steeper. Adding a constant, such as +1, shifts the entire graph upward by that amount. Understanding these transformations helps in accurately sketching the modified function.

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Stretches and Shrinks of Functions

Domain and Range of the Function

The domain of ƒ(x) = 2√x + 1 is all x ≥ 0 because the square root is undefined for negative inputs. The range is y ≥ 1 since the smallest value of 2√x is 0, and adding 1 shifts the minimum output to 1. Recognizing domain and range is essential for correct graphing.

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