Consider the following nonlinear system. Work Exercises 75 –80 in order. y = | x - 1 | y = x^2 - 4 How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

Verified step by step guidance

1. <insert step 1> Start with the basic graph of y = |x|, which is a V-shaped graph with its vertex at the origin (0,0).
2. <insert step 2> Recognize that y = |x - 1| is a horizontal transformation of y = |x|.
3. <insert step 3> To transform y = |x| into y = |x - 1|, shift the entire graph of y = |x| to the right by 1 unit.
4. <insert step 4> The vertex of the graph y = |x - 1| will now be at the point (1,0) instead of (0,0).
5. <insert step 5> The V-shape of the graph remains the same, but it is now centered at (1,0) due to the horizontal shift.

Key Concepts

Absolute Value Function

The absolute value function, denoted as y = |x|, outputs the non-negative value of x. Its graph is a V-shape that opens upwards, with the vertex at the origin (0,0). Understanding this function is crucial for analyzing transformations, as it serves as the base graph from which other variations are derived.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For the absolute value function, a horizontal shift can be applied by adjusting the input (x), while vertical shifts can be made by adding or subtracting a constant. Recognizing these transformations helps in understanding how the graph of y = |x - 1| is derived from y = |x|.

Quadratic Function

A quadratic function, represented as y = ax^2 + bx + c, produces a parabolic graph. In this context, the function y = x^2 - 4 is a downward shift of the standard parabola y = x^2 by 4 units. Understanding the properties of quadratic functions is essential for analyzing intersections and relationships with other functions, such as the absolute value function.