Problem 59 Graph each function by translati... [FREE SOLUTION]

Chapter 2: Problem 59

Graph each function by translating its parent function. $$ y=|x-1|+5 $$

Short Answer

Expert verified

Graph the parent function $ y = |x| $, then shift the graph 1 unit to the right and 5 units up. The translated function's vertex will be at (1,5).

Step by step solution

Identify the Parent Function

The parent function of the given function is the absolute value function, which is written as $ y = |x| $. The graph of the parent function consists of two rays that make a V shape, which intersects the origin (0,0).

Analyze the Transformations

To graph $ y = |x - 1| + 5 $, we need to apply transformations to the parent function $ y = |x| $. The term $ -1 $ inside the absolute value indicates a horizontal shift to the right by 1 unit, and the +5 outside of the absolute value indicates a vertical shift upwards by 5 units.

Graph the Translated Function

Start by graphing the parent function, then shift the entire graph of the parent function to the right by 1 unit and up by 5 units. The vertex of the translated absolute value function will be at the point (1,5), and the rays will extend from this vertex with the same shape as the original absolute value function.

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Key Concepts

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

Parent Function

The concept of a parent function is fundamental in understanding graph translations and transformations. A parent function is the simplest form of a function, which provides the basic outline or shape for its family of graphs. For absolute value functions, the parent function is given by the equation y = |x|. It is renowned for its characteristic 'V' shape, and its graph is a straight line making equal angles with the x-axis on both sides. The vertex of this parent function resides at the origin, which is the point (0,0).

When graphing absolute values, the parent function acts as the starting point before any shifts or alterations occur.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting a graph. There are four primary types of transformations: translations (shifts), reflections (flipping across an axis), dilations (stretching or compressing), and rotations (turning the graph).

Each transformation takes the parent graph and modifies it without changing its basic shape. For example, adding or subtracting a value from the variable x in the equation causes horizontal shifts, while adding or subtracting a value from the output y causes vertical shifts. These transformations can greatly affect the appearance and position of a graph on a coordinate plane.

Absolute Value Transformations

Absolute value transformations apply specifically to functions of the form y = |x|. When we alter this function, such as in the exercise example y = |x - 1| + 5, we're applying two transformations to the parent function:

The -1 inside the absolute value shifts the graph horizontally, to the right by 1 unit.
The +5 added to the entire function shifts the graph vertically, upwards by 5 units.

Understanding absolute value transformations is critical for graph translation, as it dictates how the 'V' shaped graph will move on the coordinate plane.

Vertex of a Function

The vertex of a function is a significant point on the graph where it changes direction. In absolute value functions, the vertex is the lowest or highest point of the 'V' shape, depending on the function's orientation on the graph. For example, in the parent function y = |x|, the vertex is at the origin (0,0).

When we apply transformations to the parent function, the vertex moves accordingly. In the case of the exercise function y=|x-1|+5, the new vertex is found at (1,5), providing a reference point to graph the transformed function properly. The position of the vertex is essential in understanding how the entire graph shifts in response to transformations.

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91影视

Graph each function by translating its parent function. $$ y=|x-1|+5 $$

Short Answer

Step by step solution

Identify the Parent Function

Analyze the Transformations

Graph the Translated Function

Key Concepts

Parent Function

Transformations of Functions

Absolute Value Transformations

Vertex of a Function

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