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.
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:
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.
TRANSFORMATIONS INVOLVING $\,\boldsymbol{y}$ (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 UP shifts $\,p\,$ units DOWN |
$(a,b+p)$ $(a,b-p)$ |
vertical translation vertical translation |
multiply by $\,-1$ | $y = -f(x)$ | reflect about $x$-axis | $(a,-b)$ | reflection about $x$-axis |
multiply by $\,g$ | $y = g\cdot f(x)$ | vertical stretch by a factor of $\,g$ |
$(a,gb)$ | vertical stretch/elongation |
divide by $\,g$ | $\displaystyle y = \frac{f(x)}{g}$ | vertical shrink by a factor of $\,g$ |
$\displaystyle \bigl(a,\frac{b}{g}\bigr)$ | vertical shrink/compression |
take absolute value | $y = |f(x)|$ | part below $x$-axis flips up | $(a,|b|)$ | absolute value |
TRANSFORMATIONS INVOLVING $\,\boldsymbol{x}$ (note that transformations involving $\,x\,$ are counter-intuitive) |
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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 LEFT shifts $\,p\,$ units RIGHT |
$(a-p,b)$ $(a+p,b)$ |
horizontal translation horizontal 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 shrink by a factor of $\,g$ |
$\displaystyle \bigl(\frac{a}{g},b\bigr)$ | horizontal shrink/compression |
every $\,x\,$ by $\,\displaystyle \frac{x}{g}\,$ | $\displaystyle y = f\bigl(\frac{x}{g}\bigr)$ | horizontal stretch by a factor of $\,g$ |
$(ga,b)$ | horizontal stretch/elongation |
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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