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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:

• shifting graphs up/down/left/right
• reflecting about axes, and the absolute value transformation
• horizontal and vertical stretching/shrinking

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:

• Need the graph of $\,y = -\sqrt{3x} + 2\,$? No problem!
  Start with the graph of $\,y = \sqrt{x}\,$ (one of the basic models):
  A) divide all the $\,x\,$-values by $\,3\,$ (a horizontal compression/shrink—the points get closer to the $y$-axis);
  B) then reflect about the $x$-axis;
  C) then move it up $2$
  You'll understand why the order BAC would also work, but the order CAB doesn't!
• 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 $\,\boldsymbol{y}$ (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 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)
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

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