Y=-F(X) Graph Ing Y= – What Does Y= F(X) Actually Mean

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

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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)
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

See more: Fluid Mechanics Fundamentals And Applications 4Th Edition, Fluid Mechanics: Fundamentals And Applications

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

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