Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=2√x+1

Verified step by step guidance

1. Identify the basic function: The function $f(x)=2\sqrt{x+1}$ is a transformation of the basic square root function $\sqrt{x}$.
2. Determine the transformations: The expression $x+1$ inside the square root indicates a horizontal shift to the left by 1 unit. The coefficient 2 outside the square root indicates a vertical stretch by a factor of 2.
3. Find the domain: Since the square root function is only defined for non-negative numbers, set the expression inside the square root greater than or equal to zero: $x+1\ge 0$. Solve for $x$ to find the domain: $x\ge -1$.
4. Identify key points: Start with the vertex of the basic function, which is at $(0,0)$. Apply the transformations to find the new vertex: $(-1,0)$. Calculate additional points by choosing values of $x$ greater than or equal to -1 and finding corresponding $y$ values.
5. Sketch the graph: Plot the vertex and additional points on the coordinate plane. Draw a smooth curve starting from the vertex and extending to the right, reflecting the shape of the square root function with the applied transformations.

Key Concepts

Function Definition

A function is a relation that assigns exactly one output for each input from its domain. In the case of ƒ(x) = 2√x + 1, the function takes a non-negative input x, applies the square root, scales it by 2, and then shifts the result up by 1. Understanding the definition of a function is crucial for graphing, as it helps identify the relationship between x and ƒ(x).

Graphing Techniques

Graphing techniques involve methods for visually representing functions on a coordinate plane. This includes identifying key features such as intercepts, asymptotes, and the overall shape of the graph. For the function ƒ(x) = 2√x + 1, recognizing that it is a transformation of the basic square root function will aid in accurately plotting its graph.

Transformations of Functions

Transformations of functions refer to changes made to the basic form of a function that affect its graph. These can include vertical shifts, horizontal shifts, stretches, and compressions. In ƒ(x) = 2√x + 1, the factor of 2 indicates a vertical stretch, while the +1 indicates a vertical shift upwards, both of which are essential for accurately graphing the function.