Curve Shifting because of Addition & Subtraction

Two Shifts
Presented through An Example:
y = (x-2)² - 4 The squaring function, y = x², drawn here in black, is a parabola. It determines what is done to numbers and therefore the shape of the curve. The function in brown, named only by the equation y = (x -2)² - 4, is also a parabola, but is shifted from x² in two ways by two components of the function. |

"Domain Shift" AgainBut WHY does a "DECREASE in the x-value cause the function, and therefore the curve, to move to the RIGHT?" In the function y=(x-2)², x² is shifted to the right. In the function y=(x+2)², x² is shifted to the left. All this happens because the value that is squared -- the value the squaring function uses as its input or domain -- is what determines the value that is output, the function value, the y-value. Recall that a function is a really dependable rule. Look above and see that when you square 0, you get 0, when you square -1 you get 1, when you square 1 you get 1, when you square 2 you get 4, and so on -- the usual ..., 9, 4, 1, 0, 1, 4, 9, ... pattern in the output or function or y-values. Look below and see how INCREASING the x by 2 before squaring moves the numbers to be squared to LEFT and DECREASING the x by 2 before squaring moves the numbers to be squared to the RIGHT. Look for the zeros to see how clearly this shift is visible. Shortly, the last long animation on this page shows the composition and how the domain and the range shifts work. Understanding composition of functions is really important. |

"Both Shift" Restated with More Detaily = (x - 2)² - 4 The graph of y = (x - 2)² - 4 depicts a function which takes a number, decreases it by 2, squares that, decreases that by 4. There is a "domain shift" forced by the x - 2. It positions the curve to the right 2 units. Notice how the axis of symmetry has changed from x=0 for x² to x=2 for (x - 2)². There is a "range shift" forced by the - 4. It positions the curve down 4 units. RELOAD the page with the browser to start the animation from the beginning. Again: In the curve y = (x - 2)² - 4, the squaring, symbolized by 1()² gives the curve its shape and dilation. The (x-2) "recenters" the curve horizontally from a axis of symmetry of x=0 to an axis of symmetry of x-2=0 or x=2. The - 4 "shifts" the curve down 4. Here, one last time in animation format, is the graphing of y = (x - 2)² - 4 from a composition of functions standpoint. |

More StuffVisit Graphing Quadratics for more examples and problems on curve shifting. For more on how functions are created see Composition of Functions. |

© 2007, Agnes Azzolino www.mathnstuff.com/math/spoken/here/2class/300/fx/shift.htm |