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Chapter 2: Problem 1

For each function, graph the function by translating the parent function. \(y=|x|-3\)

Short Answer

Expert verified
To graph the function \(y = |x| - 3\), translate the graph of the parent function \(y = |x|\) downward by 3 units, resulting in the vertex at \((0, -3)\).

Step by step solution

01

Identify the parent function

The parent function is the absolute value function, represented as: \(y = |x|\). This function forms a 'V' shape, with its vertex at the origin \((0, 0)\), and it opens upwards.
02

Identify the transformation

The given function \(y = |x| - 3\) is the parent function translated vertically downward by 3 units. There are no horizontal translations, stretches, or reflections in this case.
03

Graph the transformed function

To graph the function, take the graph of the parent function \(y = |x|\) and shift it down 3 units. This will place the vertex of the 'V' shape at the point \((0, -3)\). Maintain the shape and slope of the 'V' as in the parent function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
The absolute value function is a foundational concept in algebra and graphing. It's defined as: ewline.
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Translating Functions
When dealing with graphing functions, the concept of translating functions plays a critical role in understanding how graphs are manipulated on a coordinate system. Translation is essentially sliding the entire graph of a function horizontally or vertically without changing its shape, size, or orientation. For example, if we translate the function horizontally to the right by 2 units, every point ewline.
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Function Transformation
Function transformation involves changing the graph's appearance without altering the input-output relationship fundamental to the function's definition. There are several types of transformations, including translations that we discussed earlier, reflections, which flip the graph over a specific line like the x-axis or y-axis, and more. A reflection, for instance, reverses the orientation. ewline.
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Vertex of a Function
The vertex of a function is a significant point on the graph that helps define its shape and position. In the case of the absolute value function, the vertex is the sharpest point of the 'V' shape and serves as a pivot for any transformations applied. It is central in parabolic graphs as well, like those of quadratic functions, where the vertex is either the highest or lowest point, depending on the direction the parabola opens.

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