Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)
4 - x² |4 - x²|

검증된 단계별 안내

Identify the functions: ƒ₁(x) = 4 - x² and ƒ₂(x) = |4 - x²|. The first function is a downward-opening parabola, while the second function is the absolute value of the first.

Graph ƒ₁(x) = 4 - x²: This is a parabola that opens downwards with its vertex at (0, 4) and intersects the x-axis at x = -2 and x = 2.

Graph ƒ₂(x) = |4 - x²|: This graph will be the same as ƒ₁(x) for values where 4 - x² is positive or zero. For values where 4 - x² is negative, the graph will reflect the negative part of ƒ₁(x) above the x-axis.

Observe the effect of the absolute value: The absolute value function affects the graph by reflecting any portion of ƒ₁(x) that is below the x-axis to above the x-axis, creating a 'V' shape at the points where ƒ₁(x) crosses the x-axis.

Describe the transformation: The application of the absolute value function to ƒ₁(x) results in a graph that is symmetric about the x-axis, with all negative y-values of ƒ₁(x) becoming positive in ƒ₂(x).

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주요 개념

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

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values). For the functions ƒ₁(x) = 4 - x² and ƒ₂(x) = |4 - x²|, understanding their shapes is crucial. ƒ₁ is a downward-opening parabola, while ƒ₂ modifies this by reflecting any negative values of ƒ₁ above the x-axis, creating a 'V' shape.

Absolute Value Function

The absolute value function, denoted as |x|, transforms any negative input into a positive output. In the context of ƒ₂(x) = |4 - x²|, this means that wherever ƒ₁(x) is negative (i.e., below the x-axis), ƒ₂(x) will reflect those points to be positive. This alteration significantly changes the graph's appearance, ensuring all y-values are non-negative.

Effects of Transformations on Graphs

Transformations in calculus refer to changes made to the graph of a function, such as shifts, reflections, or stretches. Applying the absolute value function to ƒ₁(x) results in a reflection of the parts of the graph that fall below the x-axis. This transformation not only alters the graph's shape but also affects its intersections with the axes and overall behavior, making it essential to analyze these effects when comparing ƒ₁ and ƒ₂.

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.ƒ₁(x) ƒ₂(x)4 - x² |4 - x²|