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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 29
Chapter 3, Problem 29

Graph each function. See Examples 1 and 2. h(x)=|-(1/2)x|

Verified step by step guidance
1
Recognize that the function is given by \(h(x) = |-(\frac{1}{2})x|\). Since the absolute value of a number is always non-negative, the negative sign inside the absolute value will not affect the output. So, simplify the function to \(h(x) = |\frac{1}{2}x|\).
Rewrite the function as \(h(x) = \frac{1}{2} |x|\) because the absolute value of a product is the product of the absolute values, and \(|\frac{1}{2}| = \frac{1}{2}\).
Understand that the graph of \(h(x) = \frac{1}{2} |x|\) is a V-shaped graph opening upwards, with its vertex at the origin \((0,0)\).
To graph the function, create a table of values by choosing several values of \(x\) (both negative and positive), then calculate \(h(x)\) using \(h(x) = \frac{1}{2} |x|\). For example, for \(x = -2, -1, 0, 1, 2\), compute the corresponding \(h(x)\) values.
Plot the points from the table on the coordinate plane and connect them with two straight lines forming a V shape, with the vertex at the origin and slopes of \(\frac{1}{2}\) on the right side and \(-\frac{1}{2}\) on the left side.

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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 outputs the non-negative value of its input, meaning it makes all values positive or zero. For example, |x| equals x if x is positive or zero, and -x if x is negative. This affects the graph by reflecting any negative parts of the function above the x-axis.
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Function Composition

Linear Functions and Slope

A linear function has the form f(x) = mx + b, where m is the slope indicating the steepness and direction of the line. In h(x) = |-(1/2)x|, the inner function is linear with slope -1/2, which means the line decreases as x increases before applying the absolute value.
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Linear Inequalities

Graphing Transformations

Graphing transformations involve shifting, reflecting, stretching, or compressing the graph of a function. Here, the negative sign inside the absolute value reflects the linear function across the y-axis before the absolute value makes all outputs positive, resulting in a V-shaped graph symmetric about the y-axis.
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Intro to Transformations
Related Practice
Textbook Question

Graph each line. Give the domain and range. -3x + 6 = 0

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Textbook Question

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

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. through (-2,5) having slope -4

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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.

Textbook Question

Graph each piecewise-defined function.

f(x)={−12x2+2if x≤212xif x>2f(x) =\(\begin{cases}\)-\(\frac{1}{2}\)x^2 + 2 & \(\text{if }\) x \(\leq\) 2 \(\frac{1}{2}\)x & \(\text{if }\) x > 2\(\end{cases}\)

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