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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 27
Chapter 3, Problem 27

Graph each function. See Examples 1 and 2. ƒ(x)=-3|x|

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1
Identify the base function to graph, which is the absolute value function \(f(x) = |x|\). This function creates a 'V' shape with its vertex at the origin (0,0), opening upwards.
Understand the transformation applied to the base function. The given function is \(f(x) = -3|x|\), which means the absolute value function is first multiplied by 3, stretching it vertically by a factor of 3, and then multiplied by -1, reflecting it across the x-axis.
Plot the vertex of the graph at the origin (0,0), since the absolute value function's vertex remains unchanged by vertical stretching or reflection.
Choose several x-values (both positive and negative), calculate the corresponding y-values using \(f(x) = -3|x|\), and plot these points. For example, for \(x=1\), \(f(1) = -3|1| = -3\), and for \(x=-1\), \(f(-1) = -3|-1| = -3\).
Draw the graph by connecting the plotted points with straight lines forming a 'V' shape that opens downward due to the negative sign, and is steeper than the basic \(|x|\) graph because of the factor 3.

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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 |x|, outputs the distance of x from zero on the number line, always yielding a non-negative result. Its graph is a V-shaped curve with the vertex at the origin, reflecting all negative inputs to positive outputs.
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Function Composition

Vertical Stretch and Reflection

Multiplying a function by a negative constant, like -3, reflects the graph across the x-axis and stretches it vertically by a factor of 3. This changes the shape and orientation of the graph, flipping it upside down and making it steeper.
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Stretches & Shrinks of Functions

Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise linear functions, defining different expressions for x ≥ 0 and x < 0. Understanding this helps in plotting the graph accurately by considering each piece separately.
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Graphs of Logarithmic Functions
Related Practice
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Graph each line. Give the domain and range. -3x + 6 = 0

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Graph each piecewise-defined function.

f(x)={−2xif x<−33x−1if −3≤x≤2−4xif x>2f(x) =\(\begin{cases}\)-2x & \(\text{if }\) x < -3 \\3x - 1 & \(\text{if }\) -3 \(\leq\) x \(\leq\) 2 \\-4x & \(\text{if }\) x > 2\(\end{cases}\)

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Textbook Question

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

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Textbook Question

Determine whether each relation defines a function, and give the domain and range.

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Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. m=5, b=15