Sketch the graph of the function, using the curve-sketching guide of this section. f(x)=√(x^2+5)

Name: Problem 46, Chapter 4: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-08-18T19:08:32-08:00
Duration: 1 min 3 s
Channel: Carson Merrill
Description: Sketch the graph of the function, using the curve-sketching guide of this section. f(x)=√(x^2+5)

step 1 For the following problem, we want to sketch the graph of the function using the curve sketching

Sketch the graph of the function, using the curve-sketching...

Key Concepts

Domain and Range

Determining the domain and range is fundamental in curve sketching. The domain defines all the x-values for which the function is defined, while the range specifies the possible y-values. For functions involving square roots, ensuring the expression under the root is non-negative is crucial, thereby influencing both domain and range considerations.

Symmetry

Identifying symmetry in a function helps simplify the graphing process. If a function is even (i.e., f(?x) = f(x)) or odd (i.e., f(?x) = ?f(x)), you can use this property to reduce the amount of work needed to sketch the graph. Symmetry provides insights into the shape and behavior of the graph about the y-axis or the origin.

Critical Points and Local Extrema

Critical points, obtained by setting the first derivative equal to zero or locating points where the derivative does not exist, are used to determine local maximum and minimum values. These points indicate where the function may change its increasing or decreasing behavior and help in accurately outlining the shape of the graph.

First Derivative and Monotonicity

Analyzing the first derivative of the function reveals the intervals where the function is increasing or decreasing. This step is essential in determining the overall behavior of the function and in the appropriate placement of curves, peaks, and valleys on the graph.

Asymptotic Behavior

Examining the end behavior of a function helps identify any asymptotes or trends that the graph approaches as x tends to positive or negative infinity. Understanding the asymptotic behavior provides a global perspective on the graph’s shape and ensures that the sketch accurately reflects long-term tendencies.

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