Graph each function. See Examples 1 and 2. h(x) = |-½ x|

검증된 단계별 안내

Recognize that the function is given by \(h(x) = | -\frac{1}{2} x |\). The absolute value means the output is always non-negative, regardless of the sign inside the absolute value.

Simplify the expression inside the absolute value. Since \(| -a | = |a|\), rewrite the function as \(h(x) = \left| -\frac{1}{2} x \right| = \frac{1}{2} |x|\).

Understand the basic shape of \(h(x) = \frac{1}{2} |x|\). This is a V-shaped graph with its vertex at the origin \((0,0)\), opening upwards.

Plot key points by choosing values of \(x\) (for example, \(x = -2, -1, 0, 1, 2\)) and calculating \(h(x) = \frac{1}{2} |x|\). This will help you sketch the graph accurately.

Draw the graph by connecting the points with two straight lines forming a V shape, with the vertex at the origin and slopes of \(\frac{1}{2}\) for \(x > 0\) and \(-\frac{1}{2}\) for \(x < 0\).

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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 magnitude of a number or expression. Graphically, it creates a 'V' shape because all negative inputs are reflected as positive outputs, ensuring the function's values are always zero or positive.

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) = |-½ x|, the inner function is linear with slope -½, which affects the rate of change before applying the absolute value.

Graphing Transformations

Graphing transformations involve shifting, reflecting, stretching, or compressing the base graph. Here, the negative slope inside the absolute value reflects the line across the y-axis before the absolute value makes all outputs positive, resulting in a 'V' shaped graph with specific steepness.