Consider the following nonlinear system. Work Exercises 75 –80 in order. y=∣x−1∣y = | x - 1 | y=x2−4y = $$x^2 - 4$$How is the graph of y = $$x^2 - 4 $$obtained by transforming the graph of y=x2y = $$x^2$$?

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

문제 76

3장, 문제 76

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^2 - 4 obtained by transforming the graph of $y = x^{2}$ ?

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Recall the parent function for the quadratic is given by $y = x^2$, which is a parabola with vertex at the origin $(0,0)$.

The given function is $y = x^2 - 4$. Notice that this is the parent function $y = x^2$ with a constant subtracted.

Subtracting 4 from $x^2$ means every $y$-value of the original parabola is decreased by 4 units.

This results in a vertical shift of the graph downward by 4 units.

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 $(0,-4)$.

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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.

Consider the following nonlinear system. Work Exercises 75 –80 in order. y=∣x−1∣y = | x - 1 | y=x2−4y = x^2 - 4How is the graph of y = x^2 - 4 obtained by transforming the graph of y=x2y = x^2?