Lesson Constructing a function based on its given properties
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<H2>Constructing a function based on its given properies</H2> <H3>Problem 1</H3>Determine a possible equation of a exponential function that satisfies the following properties. a) Its y-intercept is 2; b) It has a horizontal asymptote at y = 5; c) it is always increasing. <B>Solution</B> <pre> y = ab^(-x) + c Coefficient "a" must be negative (I will take it in the form a = - A, where A is positive). The base "b" must be positive, and can be any positive number; I will take b = 2. Coefficient "c" must be equal to 5. y = -Ab^(-x) + 5. An additional condition is -Ab^0 + 5 = 2, which gives A = 5 - 2 = 3. Final function y = -3*2^(-x) + 5. <U>ANSWER</U> Visual check {{{graph( 400, 400, -5, 5, -5, 10, -3*2^(-x) + 5 )}}} Plot y = -3*2^(-x) + 5 </pre> <H3>Problem 2</H3>The parabola y = 2x^2 is translated to a new parabola with x intercepts 4 and -3. Find y-intercept of the new parabola. <B>Solution</B> <pre> Translations of a parabola do not change the coefficient at x^2. From the other side, the symmetry line of the new parabola is x = {{{(4 + (-3))/2}}} = 0.5 Therefore, the new parabola is y = 2*(x-0.5)^2 + b, where b is an unknown value. To find "b", use the condition that x-intercept is 4: y = 0 = 2*(x-0.5)^2 + b at x= 4, or 0 = 2*(4-0.5)^2 + b, 0 = 2*3.5^2 + b 0 = 24.5 + b b = - 24.5. Thus the new parabola is y = 2*(x-0.5)^2 - 24.5, and its value at x= 0 is y = 2*(0-0.5)^2 - 24.5 = 2*0.5^2 - 24.5 = -24. <U>ANSWER</U> </pre> Solved. Another, even more simple and short straightforward solution is possible. <pre> Since the new parabola has x-intercepts 4 and -3, the new quadratic function has the form y = a*(x+3)*(x-4) With some real coefficient "a". Since translations leave the leading coefficient at x^2 unchangeable, a = 2. It implies that the new quadratic function is y = 2*(x+3)*(x-4). Therefore, y-intercept of the new parabola is y(0) = 2*(0+3)*(0-4) = 2*3*(-4) = -24. <U>ANSWER</U> </pre> Solved (by another way). My other lessons in this site on plotting and analyzing functions are - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Finding-x-intercepts-and-y-intercepts.lesson>Finding x-intercepts and y-intercepts</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Compressing-and-stretching-of-graphs.lesson>Compressing and stretching graphs</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-PLOT-transformed-functions.lesson>HOW TO PLOT transformed functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-write-functions-for-transformed-plots.lesson>HOW TO write functions for transformed plots</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-PLOT-transformed-periodic-trigonometry-functions.lesson>HOW TO PLOT transformed periodic trigonometry functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Analyzing-periodic-trig-functions-for-amplitude-period-vert-and-hor-shifts.lesson>Analyzing periodic trigonometric functions for the amplitude, the period, vertical and horizontal shifts</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Do-not-fall-into-a-TRAP-when-analysing-problems-on-trigonometric-functions.lesson>Do not fall into a TRAP when analyzing problems on trigonometric functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/The-domain-and-the-range-of-transformed-functions.lesson>The domain and the range of transformed functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Write-a-function-that-has-given-transformations-from-the-parent-function.lesson>Write a function which is a result of given transformations of the parent function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Describe-transformations-from-the-given-basic-function-to-final-function.lesson>Describe transformations from the given parent function to final function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Writing-a-function-rule-for-a-function-based-on-its-wording-description.lesson>Writing a function rule for a function based on its wording description</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Finding-inverse-functions.lesson>Finding inverse functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Miscellaneous-problems-on-plots-of-functions.lesson>Miscellaneous problems on plots of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Given-a-point-on-a-plot-of-a-function--find-the-corresponding-point-on-the-plot-of-transformed-function.lesson>Given a point on a plot of a function, find the corresponding point on the plot of transformed function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Special-advanced-problems-on-finding-the-domain-of-functions.lesson>Special advanced problems on finding the domain of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Special-advanced-problems-on-finding-the-range-of-functions.lesson>Special advanced problems on finding the range of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/OVERVIEW-of-lessons-on-plotting-and-analyzing-functions.lesson>OVERVIEW of lessons on plotting and analyzing functions</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
