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

검증된 단계별 안내

Recognize that the function ƒ(x) = 3|x| is an absolute value function multiplied by 3. The basic shape of |x| is a 'V' shape with its vertex at the origin (0,0).

Understand that the coefficient 3 outside the absolute value stretches the graph vertically by a factor of 3. This means for every x-value, the y-value will be three times larger than the basic |x| function.

Identify key points to plot: start with the vertex at (0,0), then choose points on the right side such as (1, 3|1|) = (1,3) and (2, 3|2|) = (2,6). Similarly, for the left side, use (-1, 3|-1|) = (-1,3) and (-2, 3|-2|) = (-2,6).

Plot these points on the coordinate plane and draw straight lines connecting them, forming a 'V' shape that opens upwards with the vertex at the origin.

Label the graph clearly, noting that the function is symmetric about the y-axis because of the absolute value, and the slope of each arm of the 'V' is 3 and -3 respectively.

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주요 개념

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

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x, making all inputs positive or zero. Its graph is V-shaped, reflecting negative inputs over the y-axis. Understanding this helps in visualizing how |x| behaves and affects transformations.

Vertical Stretching of Functions

Multiplying a function by a constant greater than 1, like 3 in 3|x|, vertically stretches the graph, making it steeper. This means each output value is scaled by that factor, increasing the distance from the x-axis while preserving the shape.

Graphing Piecewise Functions

Since |x| can be expressed as a piecewise function (x if x≥0, -x if x<0), graphing involves plotting two linear parts. Recognizing this helps in accurately sketching the graph of 3|x| by applying the vertical stretch to each piece.