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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.