Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics §G TranslateRational Plots Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

MTH 55 G Review § • Any QUESTIONS About • §G → Graphing Rational Functions • Any QUESTIONS About HomeWork • §G → HW-22

GRAPH BY PLOTTING POINTS • Step1. Make a representative T-table of solutions of the equation. • Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane. • Step 3. Connect the solutions (dots) in Step 2 by a smooth curve

Translation of Graphs • Graph y = f(x) = x2. • Make T-Table & Connect-Dots • Select integers for x, starting with −2 and ending with +2. The T-table:

Translation of Graphs • Now Plot the Five Points and connect them with a smooth Curve (−2,4) (2,4) (−1,1) (1,1) (0,0)

Axes Translation • Now Move UP the graph of y = x2 by two units as shown (−2,6) (2,6) (−2,4) (2,4) (1,3) (−1,3) • What is the Equation for the new Curve? (0,2) (−1,1) (1,1) (0,0)

Axes Translation • Compare ordered pairs on the graph of with the corresponding ordered pairs on the new curve:

Axes Translation • Notice that the x-coordinates on the new curve are the same, but the y-coordinates are 2 units greater • So every point on the new curve makes the equation y = x2+2 true, and every point off the new curve makes the equation y = x2+2 false. • An equation for the new curve is thus y = x2+2

Axes Translation • Similarly, if every point on the graph of y = x2 were is moved 2 units down, an equation of the new curve is y = x2−2

Axes Translation • When every point on a graph is moved up or down by a given number of units, the new graph is called a vertical translation of the original graph.

Vertical Translation • Given the Graph of y = f(x), and c > 0 • The graph of y = f(x) +c is a vertical translation c-units UP • The graph of y = f(x) −c is a vertical translation c-units DOWN

Horizontal Translation • What if every point on the graph of y = x2 were moved 5 units to the right as shown below. (−2,4) (2,4) (3,4) (7,4) • What is the eqn of the new curve?

Horizontal Translation • Compare ordered pairs on the graph of with the corresponding ordered pairs on the new curve:

Horizontal Translation • Notice that the y-coordinates on the new curve are the same, but the x-coordinates are 5 units greater. • Does every point on the new curve make the equation y = (x+5)2 true? • No; for example if we input (5,0) we get 0 = (5+5)2, which is false. • But if we input (5,0) into the equation y = (x−5)2 , we get 0 = (5−5)2 , which is TRUE. • In fact, every point on the new curve makes the equation y = (x−5)2 true, and every point off the new curve makes the equation y = (x−5)2 false. Thus an equation for the new curve is y = (x−5)2

Horizontal Translation • Given the Graph of y = f(x), and c > 0 • The graph of y = f(x−c) is a horizontal translation c-units to the RIGHT • The graph of y = f(x+c) is a horizontal translation c-units to the LEFT

Example  Plot by Translation • Use Translation to graph f(x) = (x−3)2−2 • LET y = f(x) → y = (x−3)2−2 • Notice that the graph of y = (x−3)2−2 has the same shape as y = x2, but is translated 3-unit RIGHT and 2-units DOWN. • In the y = (x−3)2−2, call −3 and −2translators

Example  Plot by Translation • The graphs of y=x2 and y=(x−3)2−2 are different; although they have the Same shape they have different locations • Now remove the translators by a substitution of x’ (“x-prime”) for x, and y’ (“y-prime”) for y • Remove translators for an (x’,y’) eqn

Example  Plot by Translation • Since the graph of y=(x−3)2−2 has the same shape as the graph of y’ =(x’)2 we can use ordered pairs of y’ =(x’)2 to determine the shape • T-tablefory’ =(x’)2

Example  Plot by Translation • Next use the translation rules to find the origin of the x’y’-plane. Draw the x’-axis and y’-axis through the translated origin • The origin of the x’y’-plane is 3 units right and 2 units down from the origin of the xy-plane. • Through the translated origin, we use dashed lines to draw a new horizontal axis (the x’-axis) and a new vertical axis (the y’-axis).

Example  Plot by Translation • Locate the Origin of the Translated Axes Set using the translator values Move: 3-Right, 2-Down

Example  Plot by Translation • Now Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph. • Remember that this graph is smooth

Example  Plot by Translation • Perform a partial-check to determine correctness of the last graph. Pick any point on the graph and find its (x,y) CoOrds; e.g., (4, −1) is on the graph • The Ordered Pair (4, −1) should make the xy Eqn True

Example  Plot by Translation • Sub (4, −1) into y=(x−3)2−2 : • Thus (4, −1) does make y = (x−3)2−2 true. In fact, all the points on the translated graph make the original Eqn true, and all the points off the translated graph make the original Eqn false

Example  Plot by Translation • What are the Domain &Range of y=(x−3)2−2? • To find the domain & range of the xy-eqn, examine the xy-graph (not the x’y’ graph). • The xy graph showns • Domain of f is {x|x is any real number} • Range of f is {y|y ≥ −2}

Graphing Using Translation • Let y = f(x) • Remove the x-value & y-value “translators” to form an x’y’ eqn. • Find ordered pair solutions of the x’y’ eqn • Use the translation rules to find the origin of the x’y’-plane. Draw dashed x’ and y’ axes through the translated origin. • Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.

Example ReCallGraph y = |x| • Make T-table

Example Graph y = |x+2|+3 • Step-1 • Step-2 • Step-3 → T-table in x’y’

Example Graph y = |x+2|+3 • Step-4: the x’y’-plane origin is 2 units LEFT and 3 units UP from xy-plane Left 2 Up 3

Example Graph y = |x+2|+3 • Step-5: Remember that the graph of y = |x| is V-Shaped: Left 2 Up 3

Rational Function Translation • A rational function is a function f that is a quotient of two polynomials, that is, • Where • where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. • The domain of f consists of all inputs xfor which q(x) ≠ 0.

Example  • Find the DOMAIN and GRAPH for f(x) • SOLUTIONWhen the denom x = 0, we have x = 0, so the only input that results in a denominator of 0 is 0. Thus the domain {x|x  0} or (–, 0)  (0, ) • Construct T-table • Next Plot points & connect Dots

Plot • Note that the Plot approaches, but never touches, • the y-axis (as x ≠ 0) • In other words the graph approaches the LINE x = 0 • the x-axis (as 1/  0) • In other words the graph approaches the LINE y = 0 • A line that is approached by a graph is called an ASYMPTOTE

ReCall Asymptotic Behavior • The graph of a rational function never crosses a vertical asymptote • The graph of a rational function might cross a horizontal asymptote but does not necessarily do so

y = 3x2 Recall Vertical Translation • Given the graph of the equation y = f(x), and c > 0, • the graph of y = f(x) +cis the graph of y = f(x) shifted UP(vertically) c units; • the graph of y = f(x) –cis the graph of y = f(x) shifted DOWN(vertically) c units y = 3x2+2 y = 3x2−3

y = 3x2 Recall Horizontal Translation • Given the graph of the equation y = f(x), and c > 0, • the graph of y = f(x–c) is the graph of y = f(x) shifted RIGHT(Horizontally) c units; • the graph of y = f(x+c) is the graph of y = f(x) shifted LEFT(Horizontally) c units. y = 3(x-2)2 y = 3(x+2)2

ReCall Graphing by Translation • Let y = f(x) • Remove the translators to form an x’y’ eqn • Find ordered pair solutions of the x’y’ eqn • Use the translation rules to find the origin of the x’y’-plane. Draw the dashed x’ and y’ axes through the translated origin. • Plot the ordered pairs of the x’y’ equation on the x’y’-plane, and use the points to draw an appropriate graph.

Example  Graph • Step-1 • Step-2 • Step-3 → T-table in x’y’

Example  Graph • Step-4: The origin of the x’y’ -plane is 2 units left and 1 unit up from the origin of the xy-plane:

Example  Graph • Step-5: We know that the basic shape of this graph is Hyperbolic. Thus we can sketch the graph using Fewer Points on the translated axis using the T-Table

Example  Graph • Examination of the Graph reveals • Domain →{x|x≠ −2} • Range →{y|y≠ 1}

Example  Graph • Step-1 • Step-2 • Step-3 → T-table in (x’y’) by y’ = −2/x’

Example  Graph • Step-4: The origin of the x’y’ -plane is 3 units RIGHT and 1 unit DOWN from the origin of the xy-plane

Example  Graph • Notice for this Graph that the Hyperbola is • “mirrored”, or rotated 90°, by the leading Negative sign • “Spread out”, or expanded, by the 2 in the numerator

Example  Graph • Examination of the Graph reveals • Domain →{x|x≠ 3} • Range →{y|y≠ −1}

Example  Graph • Step-1 • Step-2 • Step-3 → T-table in x’y’ by y’ = −1/(2x’)

Example  Graph • Step-4: The origin of the x’y’ -plane is 1 unit RIGHT and 2 units UP from the origin of the xy-plane

Example  Graph • Notice for this Graph that the Hyperbola is • “mirrored”, or rotated 90°, by the leading Negative sign • “Pulled in”, or contracted, by the 2 in the Denominator

Example  Graph • Examination of the Graph reveals • Domain →{x|x≠ 1} • Range →{y|y≠ 2}

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege