LESSON READ-THROUGH** by Dr. Carol JVF Burns (website creator) Follow along with the highlighted text while you listen! **

You are watching: Y=-f(x) graph

The lesson Graphs of Functions in the Algebra II curriculum gives a thorough introduction to graphs of functions. For those who need only a quick review, the key concepts are repeated here. The exercises in this lesson duplicate those in Graphs of Functions.

The equation ‘$,y = f(x),$’ is an equation in two variables, $,x,$ and $,y,$. The *graph* of the equation $,y = f(x),$ is the picture of all the points $,(x,y),$ that make it true; observe that to make this equation true, $,y,$ must equal $,f(x),$. Thus, the graph of the equation $,y = f(x),$ is the set of all points of the form $,color{green}{igl(x,overbrace{f(x)}^{y}igr)},$. Compare! The following two requests ask for exactly the same thing: — Graph the equation $,color{green}{y = f(x)},$. — Graph the function $,color{green}{f},$. Both are asking for the set of all points of the form $,igl(x,f(x)igr),$.

## Understanding the Relationship between an Equation and its Graph

There are things that you can *DO* to an equation $,y = f(x),$ that will change its graph. Or, there are things that you can *DO* to a graph that will change its equation. Stretching, shrinking, moving up/down/left/right, reflecting about axes; they”re all covered thoroughly in the next few web exercises:

An understanding of these graphical transformations makes it easy to graph a wide variety of functions, by starting with a basic model and then applying a sequence of transformations to change it to the desired function.

For example, *after mastering the graphical transformations*, you”ll be able to do the following: Or, suppose you have the graph (call it $,G,$) of a known equation $,y = f(x),$. The graph is being changed, though, and you need the corresponding new equations. — If $,G,$ is shifted up $,2,$ units, the new equation is $,y = f(x) + 2,$. (Transformations involving $,y,$ are intuitive.) — If $,G,$ is shifted right $,2,$ units, the new equation is $,y = f(x-2),$. (Transformations involving $,x,$ are counter-intuitive.) — If $,G,$ is reflected about the $y$-axis, the new equation is $,y = f(-x),$.

For your convenience, all the graphical transformations are summarized in the GRAPHICAL TRANSFORMATIONS table below. Given *any* entry in a row, you should (eventually!) be able to fill in all the remaining entries in that row.

** SUMMARY: GRAPHICAL TRANSFORMATIONS SET-UP FOR THE TABLE:**** you”re starting with the equation $y = f(x)$ (so, the ‘previous $color{purple}{y}$-value’—see the first column below—is $,f(x)$) assume: $p > 0$ (‘$p$’ for Positive) $g > 1$ (‘$g$’ for Greater than) the point $,(a,b),$ is a point on the graph of $,y = f(x),$, so that the equation $,b = f(a),$ is true **

TRANSFORMATIONS INVOLVING $,oldsymbol{y}$ (note that transformations involving $,y,$ are intuitive) |
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DO THIS TO THE PREVIOUS $y$-VALUE | NEW EQUATION | NEW GRAPH | $(a,b)$ MOVES TO … | TRANSFORMATION TYPE |

add $,p$subtract $,p$ | $y = f(x) + p$$y = f(x) – p$ | shifts $,p,$ units UPshifts $,p,$ units DOWN | $(a,b+p)$$(a,b-p)$ | vertical translationvertical translation |

multiply by $,-1$ | $y = -f(x)$ | reflect about $x$-axis | $(a,-b)$ | reflection about $x$-axis |

multiply by $,g$ | $y = gcdot f(x)$ | vertical stretchby a factor of $,g$ | $(a,gb)$ | vertical stretch/elongation |

divide by $,g$ | $displaystyle y = frac{f(x)}{g}$ | vertical shrinkby a factor of $,g$ | $displaystyle igl(a,frac{b}{g}igr)$ | vertical shrink/compression |

take absolute value | $y = |f(x)|$ | part below $x$-axis flips up | $(a,|b|)$ | absolute value |

TRANSFORMATIONS INVOLVING $,oldsymbol{x}$ (note that transformations involving $,x,$ are counter-intuitive) | ||||

REPLACE … | NEW EQUATION | NEW GRAPH | $(a,b)$ MOVES TO … | TRANSFORMATION TYPE |

every $,x,$ by $,x+p,$every $,x,$ by $,x-p$ | $y = f(x+p)$$y = f(x-p)$ | shifts $,p,$ units LEFTshifts $,p,$ units RIGHT | $(a-p,b)$$(a+p,b)$ | horizontal translationhorizontal translation |

every $,x,$ by $,-x,$ | $y = f(-x)$ | reflect about $y$-axis | $(-a,b)$ | reflection about $y$-axis |

every $,x,$ by $,gx,$ | $y = f(gx)$ | horizontal shrinkby a factor of $,g$ | $displaystyle igl(frac{a}{g},bigr)$ | horizontal shrink/compression |

every $,x,$ by $,displaystyle frac{x}{g},$ | $displaystyle y = figl(frac{x}{g}igr)$ | horizontal stretchby a factor of $,g$ | $(ga,b)$ | horizontal stretch/elongation |

Master the ideas from this section by practicing the exercise at the bottom of this page. When you”re done practicing, move on to: shifting graphs up/down/left/right

See more: How Many Gallons In A 22 X 52 Pool Set With Cartridge Filter Pump

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. | PROBLEM TYPES: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

AVAILABLE MASTERED IN PROGRESS