Question

Sketch the graph of the function in Exercise 11 .

   Sketch the graph of the function in Exercise 11 .

Calculus: A Complete Course

Calculus: A Complete Course

Robert Adams ,… 9th Edition

Chapter 2, Problem 18

Instant Answer

Solved by Expert Linda Hand

02/13/2022

Step 1

The absolute value of x is equal to x if x is positive or zero, and it's the opposite of x if x is negative. So, if we have the absolute value of 4x - 1, it's equal to 4x - 1 if 4x - 1 is positive, and it's the opposite of 4x - 1 if 4x - 1 is negative. In Show more…

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Sketch the graph of the function in Exercise 11 .

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Key Concepts

Transformation of Functions

Transformation of functions refers to operations such as translations, stretches, compressions, and reflections that alter the basic shape of a graph. Recognizing these transformations can simplify the sketching process by relating the function to a more familiar or standard form.

Intercepts

Identifying the intercepts, where the graph crosses the coordinate axes, is a fundamental step in graphing functions. The x-intercepts show where the function equals zero, and the y-intercept indicates the value of the function when the input is zero.

Asymptotic Behavior

Understanding asymptotic behavior involves analyzing how a function behaves as the input approaches certain critical values or infinity. This includes identifying horizontal, vertical, or oblique asymptotes that indicate the long-term behavior of the function.

Critical Points and Turning Points

Critical points are values in the domain where the derivative is zero or undefined, which can correspond to local maximums, minimums, or points of inflection. These points are crucial for accurately sketching the overall shape of the graph.

Graphing Functions

Graphing functions involves representing the set of all ordered pairs that satisfy the function's equation on a coordinate plane. This process requires understanding the behavior of the function across its domain and identifying key features that define its shape.

Domain and Range

The domain and range of a function indicate the set of input values for which the function is defined and the set of output values it can produce. Determining these sets is essential for understanding where the graph exists and how it behaves.

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