# 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},bigr)\$ horizontal shrink/compression every \$,x,\$ by \$,displaystyle frac{x}{g},\$ \$displaystyle y = figl(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

 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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