{"id":1956,"date":"2019-08-26T10:05:13","date_gmt":"2019-08-26T15:05:13","guid":{"rendered":"http:\/\/www.kaptest.com\/blog\/prep\/?p=1956"},"modified":"2020-09-11T20:41:14","modified_gmt":"2020-09-11T20:41:14","slug":"psat-math-function-behavior-and-transformations","status":"publish","type":"post","link":"https:\/\/wpapp.kaptest.com\/study\/psat\/psat-math-function-behavior-and-transformations\/","title":{"rendered":"PSAT Math: Function Behavior and Transformations"},"content":{"rendered":"<p>When describing the graph of a function or an interval (a specific segment) of a function, the trend of the relationship between the <i>x-<\/i> and <i>y-<\/i> values while reading the graph from left to right is often important. Three terms you are sure to see in more difficult\u00a0function questions are <b>increasing<\/b>, <b>decreasing<\/b>, and <b>constant<\/b>. Let\u2019s look at what these terms mean and how they apply to PSAT questions.<\/p>\n<ul>\n<li><b><\/b><b>Increasing<\/b> functions have <i>y-<\/i>values that<i> increase<\/i> as the corresponding <i>x-<\/i>values increase.<\/li>\n<li><b><\/b><b>Decreasing<\/b> functions have <i>y-<\/i>values that <i>decrease<\/i> as the corresponding <i>x-<\/i>values increase.<\/li>\n<li><b><\/b><b>Constant<\/b> functions have <i>y-<\/i>values that <i>stay the same<\/i> as the <i>x-<\/i>values increase.<\/li>\n<\/ul>\n<p>The PSAT can ask about function trends in a variety of ways. The most basic would be to examine a function\u2019s behavior and determine whether (and where) the function\u00a0is increasing, decreasing, or constant. Tougher questions might ask you to identify the trend and then explain what it means in the context of a real-life situation presented in the question, or to identify the effect a transformation would have on the trend of a function.<br \/>\nA function <b>transformation<\/b> occurs when a change is made to the function\u2019s equation or graph. Transformations include translations (moving a graph up\/down, left\/right), reflections (flips about an axis or other line), and expansions\/compressions (stretching or squashing horizontally or vertically). How do you know which is occurring? The following table provides some rules for guidance when altering a hypothetical function <i>f<\/i>(<i>x<\/i>).<br \/>\n&nbsp;<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-0  el_before_av_heading  avia-builder-el-first  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >PSAT Math: Transforming Functions<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><\/p>\n<table cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td valign=\"middle\"><b>Algebraic Change<\/b><\/td>\n<td valign=\"middle\"><b>Corresponding Graphical Change<\/b><\/td>\n<td valign=\"middle\"><b>Graph<\/b><\/td>\n<td valign=\"middle\"><b>Algebraic Change<\/b><\/td>\n<td valign=\"middle\"><b>Corresponding Graphical Change<\/b><\/td>\n<td valign=\"middle\"><b>Graph<\/b><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>)<\/td>\n<td valign=\"middle\">N\/A\u2014original function<\/td>\n<td valign=\"middle\">\u00a0<a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_04.png\"><img loading=\"lazy\" class=\"aligncenter  wp-image-1957\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_04.png\" alt=\"psat_c09_cb_04\" width=\"272\" height=\"232\" \/><\/a><\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i> + <i>a<\/i>)<\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) moves left <i>a<\/i> units<\/td>\n<td valign=\"middle\">\u00a0<a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_07.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1958\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_07-300x223.png\" alt=\"psat_c09_cb_07\" width=\"300\" height=\"223\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) + <i>a<\/i><\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) moves up <i>a<\/i> units<\/td>\n<td valign=\"middle\"><a style=\"font-family: inherit;font-size: inherit\" href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_05.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1961\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_05-300x223.png\" alt=\"psat_c09_cb_05\" width=\"300\" height=\"223\" \/><\/a><\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i> \u2212 <i>a<\/i>)<\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) moves right <i>a<\/i> units<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_08.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1964\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_08-300x223.png\" alt=\"psat_c09_cb_08\" width=\"300\" height=\"223\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) \u2212 <i>a<\/i><\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) moves down <i>a<\/i> units<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_06.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1963\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_06-300x223.png\" alt=\"psat_c09_cb_06\" width=\"300\" height=\"223\" \/><\/a><\/td>\n<td valign=\"middle\">\u2212\u200a<i>f<\/i>(<i>x<\/i>)<\/td>\n<td valign=\"middle\"><i>f<\/i>\u200a(<i>x<\/i>) reflected over the <i>x<\/i>-axis (top to bottom)<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_09.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1965\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_09-300x300.png\" alt=\"psat_c09_cb_09\" width=\"300\" height=\"300\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>f<\/i>(\u2212<i>x<\/i>)<\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) reflected over the <i>y<\/i>-axis (left to right)<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_10.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1966\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_10-300x300.png\" alt=\"psat_c09_cb_10\" width=\"300\" height=\"300\" \/><\/a><\/td>\n<td valign=\"middle\"><i>af<\/i>(<i>x<\/i>) (0 &lt; <i>a<\/i> &lt; 1)<\/td>\n<td valign=\"middle\"><i>f<\/i>\u200a(<i>x<\/i>) undergoes vertical compression<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_13.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1969\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_13-300x300.png\" alt=\"psat_c09_cb_13\" width=\"300\" height=\"300\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>f<\/i>(<i>ax<\/i>) (0 &lt; <i>a<\/i> &lt; 1)<\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) undergoes horizontal expansion<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_11.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1967\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_11-300x300.png\" alt=\"psat_c09_cb_11\" width=\"300\" height=\"300\" \/><\/a><\/td>\n<td valign=\"middle\"><i>af<\/i>(<i>x<\/i>)(<i>a<\/i> &gt; 1)<\/td>\n<td valign=\"middle\"><i>f<\/i>\u200a(<i>x<\/i>) undergoes vertical expansion<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_14.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1970\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_14-300x300.png\" alt=\"psat_c09_cb_14\" width=\"300\" height=\"300\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><i>f<\/i>(<i>ax<\/i>)(<i>a<\/i> &gt; 1)<\/td>\n<td valign=\"middle\"><i>f<\/i>(<i>x<\/i>) undergoes horizontal compression<\/td>\n<td valign=\"middle\"><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_12.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1968\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_12-300x300.png\" alt=\"psat_c09_cb_12\" width=\"300\" height=\"300\" \/><\/a><\/td>\n<td colspan=\"3\" valign=\"middle\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If you forget what a particular transformation looks like, you can always plug in a few values for <i>x<\/i> and plot the points to determine the effect on the function\u2019s graph.<br \/>\n<b>Expert Tip<\/b><br \/>\nIf you forget what a particular transformation looks like, you can always plug in a few values for <i>x<\/i> and plot the points to determine the effect on the function\u2019s graph.<br \/>\n<b>Expert Tip<\/b><br \/>\nAdding or subtracting inside the parentheses of a function will always cause a horizontal change (e.g., shift left\/right, horizontal reflection); if the alteration is outside the parentheses, the result is a vertical change.<br \/>\n&nbsp;<br \/>\n<b>Note<\/b><br \/>\nNote that the slope of the function post-translation is identical to that of the original function. Translations only shift a function and do not impact slope.<br \/>\n&nbsp;<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-1  el_after_av_heading  el_before_av_promobox  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >PSAT Practice Question: Transforming Functions<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nA function transformation question for you to try follows.<br \/>\n\t<div  style='background:#ffffff;color:#444444;border-color:#444444;' class='av_promobox  avia-button-no   avia-builder-el-2  el_after_av_heading  avia-builder-el-last '>\t\t<div class='avia-promocontent'><p>\n<div class=\"flex_column av_one_full  flex_column_div first  avia-builder-el-3  el_before_av_one_half  avia-builder-el-first  \" ><p><a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_15_2.png\"><img loading=\"lazy\" class=\"aligncenter  wp-image-1973\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_15_2.png\" alt=\"psat_c09_cb_15_2\" width=\"219\" height=\"223\" \/><\/a><br \/>\n1. The graph above represents the function\u00a0<i>f<\/i>\u200a(<i>x<\/i>). Which of the following choices corresponds to<span class=\"no-break\">\u00a0f(x \u2013 2) \u2013 5\u00a0?<\/span><\/p><\/div><br \/>\n<div class=\"flex_column av_one_half  flex_column_div first  avia-builder-el-4  el_after_av_one_full  el_before_av_one_half  column-top-margin\" ><p>A.<br \/>\n<div  class='avia-image-container  av-styling-    avia-builder-el-5  el_before_av_image  avia-builder-el-first  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_16_2.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div><br \/>\nB.<br \/>\n<div  class='avia-image-container  av-styling-    avia-builder-el-6  el_after_av_image  avia-builder-el-last  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_17_2.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div><\/p><\/div><br \/>\n<div class=\"flex_column av_one_half  flex_column_div   avia-builder-el-7  el_after_av_one_half  avia-builder-el-last  column-top-margin\" ><p>C.<br \/>\n<div  class='avia-image-container  av-styling-    avia-builder-el-8  el_before_av_image  avia-builder-el-first  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_18_2.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div><br \/>\nD.<br \/>\n<div  class='avia-image-container  av-styling-    avia-builder-el-9  el_after_av_image  avia-builder-el-last  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_19_2.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div><\/p><\/div><\/p>\n<\/div><\/div><br \/>\n&nbsp;<br \/>\nUse the Kaplan Method for Math to solve this question, working through it step-by-step. The following table shows Kaplan\u2019s strategic thinking on the left, along with suggested math scratchwork on the right.<br \/>\n&nbsp;<\/p>\n<table cellspacing=\"0\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td valign=\"middle\"><b>Strategic Thinking<\/b><\/td>\n<td valign=\"middle\"><b>Math Scratchwork<\/b><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><b>Step 1: Read the question, identifying and organizing important information as you go<\/b>You must determine which graph shows the transformation specified in the question stem.<\/td>\n<td valign=\"middle\"><\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><b>Step 2: Choose the best strategy to answer the question<\/b><i>How do you begin solving?<\/i>First, determine what the transformation is.Next, identify a couple of easy points on the initial function and apply the transformation \u201cinstructions\u201d to them. The <i>y<\/i>-intercept is a good choice here.<i>On which graph do the transformed points lie?<\/i>Determine which answer\u2019s graph contains the new coordinates.<\/td>\n<td valign=\"middle\">graph moves 2 units right and 5 units down(0, 1) becomes (2, \u22124)(1, 2) becomes (3, \u22123)new points fall on the graph of (B)<\/td>\n<\/tr>\n<tr>\n<td valign=\"middle\"><b>Step 3: Check that you answered the<\/b> <b><i>right<\/i><\/b> <b>question<\/b>The only matching graph is (B).<\/td>\n<td valign=\"middle\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><b>Expert Tip<\/b><br \/>\nIf you\u2019re better with algebra, plug the transformation changes into the original function instead of picking points. Substituting <i>x<\/i> \u2212 2 for <i>x<\/i> and subtracting another 5 gives <i>y<\/i> = [(<i>x<\/i> \u2212 2) + 1] \u2212 5 = <i>x<\/i> \u2212 6. The graph of <i>y<\/i> = <i>x<\/i> \u2212 6 clearly corresponds to (B).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>When describing the graph of a function or an interval (a specific segment) of a function, the trend of the relationship between the x- and y- values while reading the graph from left to right is often important. Three terms you are sure to see in more difficult\u00a0function questions are increasing, decreasing, and constant. Let\u2019s [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28736,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[240],"tags":[],"_links":{"self":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/1956"}],"collection":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/comments?post=1956"}],"version-history":[{"count":3,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/1956\/revisions"}],"predecessor-version":[{"id":34741,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/1956\/revisions\/34741"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media\/28736"}],"wp:attachment":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media?parent=1956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/categories?post=1956"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/tags?post=1956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}