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

Verified step by step guidance

1. Recall the parent function for the quadratic is given by \(y = x^2\), which is a parabola with vertex at the origin \((0,0)\).
2. The given function is \(y = x^2 - 4\). Notice that this is the parent function \(y = x^2\) with a constant subtracted.
3. Subtracting 4 from \(x^2\) means every \(y\)-value of the original parabola is decreased by 4 units.
4. This results in a vertical shift of the graph downward by 4 units.
5. Therefore, the graph of \(y = x^2 - 4\) is obtained by shifting the graph of \(y = x^2\) down 4 units, moving the vertex from \((0,0)\) to \((0,-4)\).

Key Concepts

Parent Function and Transformations

The parent function y = x^2 is the basic quadratic function whose graph is a parabola centered at the origin. Transformations involve shifting, stretching, or reflecting this graph to produce new functions. Understanding how changes to the equation affect the graph is essential for analyzing y = x^2 - 4.

Vertical Shifts

A vertical shift moves the graph up or down without changing its shape. In the function y = x^2 - 4, subtracting 4 shifts the entire parabola downward by 4 units. This means every point on y = x^2 moves 4 units lower on the y-axis.

Graphing Nonlinear Systems

Graphing nonlinear systems involves plotting multiple nonlinear equations to find points of intersection or analyze their behavior. Understanding each graph individually, such as y = |x - 1| and y = x^2 - 4, helps in comparing and solving the system.