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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.2.39
Chapter 1, Problem 1.2.39

Graph the functions in Exercises 37–56.


y = |x − 2|

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Identify the basic form of the function: The given function is \( y = |x - 2| \), which is an absolute value function. The graph of an absolute value function \( y = |x - a| \) is a V-shaped graph with its vertex at the point \( (a, 0) \).
Determine the vertex: For the function \( y = |x - 2| \), the vertex is at \( (2, 0) \). This is because the expression inside the absolute value, \( x - 2 \), equals zero when \( x = 2 \).
Analyze the slope of the lines: The graph of \( y = |x - 2| \) consists of two linear pieces. For \( x < 2 \), the function behaves like \( y = -(x - 2) = -x + 2 \), which has a negative slope. For \( x > 2 \), the function behaves like \( y = x - 2 \), which has a positive slope.
Plot key points: Start by plotting the vertex at \( (2, 0) \). Then, choose points on either side of the vertex to plot. For example, for \( x = 1 \), \( y = |1 - 2| = 1 \), and for \( x = 3 \), \( y = |3 - 2| = 1 \). Plot these points: \( (1, 1) \) and \( (3, 1) \).
Draw the graph: Connect the points with straight lines to form the V-shape. The left side of the V will have a negative slope, and the right side will have a positive slope, both meeting at the vertex \( (2, 0) \). This completes the graph of the function \( y = |x - 2| \).

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Key Concepts

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

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. For the function y = |x - 2|, it represents the distance between x and 2, creating a V-shaped graph that opens upwards with its vertex at the point (2, 0).
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Average Value of a Function

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize mathematical functions. For y = |x - 2|, one can identify key points, such as the vertex and intercepts, and use symmetry to sketch the graph accurately. Understanding how to plot these points is essential for creating a clear representation of the function.
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Graphing The Derivative

Vertex of a Function

The vertex of a function is the point where the graph changes direction, often representing a minimum or maximum value. In the case of y = |x - 2|, the vertex is located at (2, 0), which is the lowest point of the V-shaped graph. Recognizing the vertex helps in understanding the overall shape and behavior of the function.
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Properties of Parabolas
Related Practice
Textbook Question

Using the Addition Formulas


Use the addition formulas to derive the identities in Exercises 31–36.


sin (A − B) = sin A cos B − cos A sin B

Textbook Question

Shifting Graphs


The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.


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Textbook Question

Using the Half-Angle Formulas


Find the function values in Exercises 47–50.


sin² 3π/8

Textbook Question

Refer to the given figure. Write the radius r of the circle in terms of α and θ.

Textbook Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.


𝔂 = e⁻ˣ²

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Textbook Question

Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


f(x) = x² + x

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