Graph each function. See Examples 1 and 2. h(x) = √4x

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Identify the function to be graphed: $h(x) = \sqrt{4x}$. This is a square root function where the expression inside the root is \$4x$.

Determine the domain of the function. Since the square root requires the radicand to be non-negative, set $4x \geq 0$ which simplifies to $x \geq 0$. So, the function is defined for all $x$ greater than or equal to zero.

Create a table of values by choosing several $x$ values within the domain (for example, $x=0, 1, 2, 4$) and calculate the corresponding $h(x)$ values using $h(x) = \sqrt{4x}$.

Plot the points from the table on the coordinate plane. For instance, when $x=0$, $h(0) = \sqrt{0} = 0$; when $x=1$, $h(1) = \sqrt{4} = 2$, and so on.

Draw a smooth curve through the plotted points starting at the origin $(0,0)$ and increasing to the right, reflecting the shape of the square root function which grows slowly as $x$ increases.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain is the set of all input values (x) for which the function is defined. For h(x) = √4x, the expression inside the square root must be non-negative, so 4x ≥ 0, meaning x ≥ 0. Understanding the domain ensures the graph only includes valid points.

Square Root Function

The square root function, √x, outputs the non-negative number whose square is x. It is defined only for x ≥ 0 and produces a curve starting at the origin and increasing slowly. Recognizing this shape helps in graphing h(x) = √4x by scaling the input.

Function Transformation and Scaling

Multiplying the input by a constant inside the function, as in √4x, affects the graph horizontally. Specifically, √4x = √(4 * x) = 2√x, which vertically stretches the basic square root graph by a factor of 2. Understanding transformations helps in accurately sketching the graph.

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