{"id":5483,"date":"2020-09-11t11:30:51","date_gmt":"2020-09-11t18:30:51","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=5483"},"modified":"2016-11-13t15:11:43","modified_gmt":"2016-11-13t23:11:43","slug":"sat-math-heart-of-algebra","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/sat\/sat-math-heart-of-algebra\/","title":{"rendered":"sat math: heart of algebra"},"content":{"rendered":"<p><img decoding=\"async\" class=\"alignnone size-full wp-image-15777\" src=\"\/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header.jpg\" alt=\"an open notebook with algebra practice problems with a calculator on the corner and a pen resting on the notebook and a book next to it\" width=\"1200\" height=\"600\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header.jpg 2400w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header-300x150.jpg 300w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header-600x300.jpg 600w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header-768x384.jpg 768w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header-1536x768.jpg 1536w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/2400x1200-blog-header-2048x1024.jpg 2048w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p>heart of algebra is one of <a href=\"https:\/\/collegereadiness.collegeboard.org\/about\/alignment\/math\/heart-of-algebra\" target=\"_blank\" rel=\"noopener noreferrer\">3 sections<\/a> on the <a href=\"https:\/\/collegereadiness.collegeboard.org\/sat\/inside-the-test\/math\" target=\"_blank\" rel=\"noopener noreferrer\">sat math test<\/a>, and will include 19 of the total 58 questions. in short, \u201cheart of algebra\u201d refers to any problem that has an algebraic equation in which none of the powers next to a variable is higher than one. for example:<\/p>\n<p><center><span style=\"color: #26b67c;\">\\(3x + 5 = 2\\)<\/span><\/p>\n<p><span style=\"color: #26b67c;\">\\(y + 5 &lt; -1\\)<\/span><\/center><\/p>\n<p>pretty straightforward, right? the computation underpinning heart of algebra is straightforward, but the sat typically won\u2019t be giving you problems this simple. while you will encounter several problems that ask you to solve for variables, much of the heart of algebra content are real-world word problems that you have to unwrap before solving. for example:<\/p>\n<div class=\"mox\">\n<p>five lemonades and two cookies cost $1.50. two lemonades and five cookies cost $2.70. how much do one cookie and one lemonade cost?<\/p>\n<p><em>(wondering how to solve this equation? read on and you&#8217;ll understand how!)<\/em><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>how to prepare for the sat heart of algebra section<\/h2>\n<p>there are two ways to make sure you&#8217;re prepared for heart of algebra questions:<\/p>\n<ol>\n<li>understand how to solve linear equations and systems of equations, along with common variations, like inequalities.<\/li>\n<li>practice translating real-world scenarios and information into a mathematical equation.<\/li>\n<\/ol>\n<p>to help you get the swing of things, we\u2019ve broken down and demonstrated the key concepts and skills that will serve you in the heart of algebra section. feel free to explore these concepts through the clickable table of contents below:<\/p>\n<div class=\"mox\">\n<ul>\n<li><a href=\"#algebrabasics\">algebra basics: balancing the equation<\/a>\n<ul>\n<li><a href=\"#inequalities\">inequalities<\/a><\/li>\n<li><a href=\"#systemofeq\">system of equations<\/a><\/li>\n<li><a href=\"#algebrapractice\">algebra basics practice: system of equations<\/a><\/li>\n<\/ul>\n<\/li>\n<li><a href=\"#coordinategeom\">coordinate geometry basics<\/a>\n<ul>\n<li><a href=\"#quadrants\">quadrants<\/a><\/li>\n<li><a href=\"#axes\">the x-axis &amp; y-axis<\/a><\/li>\n<li><a href=\"#integerscale\">integers and scale of graphs<\/a><\/li>\n<li><a href=\"#coordinates\">coordinates<\/a><\/li>\n<li><a href=\"#slope\">slope<\/a><\/li>\n<li><a href=\"#coordinatepractice\">coordinate geometry practice<\/a><\/li>\n<\/ul>\n<\/li>\n<li><a href=\"#functions\">functions<\/a>\n<ul>\n<li><a href=\"#gofxexplained\">the \u201cg\u201d in g(x) is just shorthand<\/a><\/li>\n<li><a href=\"#nestedfunctions\">functions inside of functions<\/a><\/li>\n<li><a href=\"#functionspractice\">functions practice: nested functions<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>algebra basics: balancing the equation<\/h2>\n<p>first thing\u2019s first: in order to interpret real-world scenarios as equations, you\u2019ll need to be super clear on how to balance an equation!<\/p>\n<p>so, in an algebraic equation with an equal sign, think of the equation in balance. if you do anything to one side you do the exact same thing to the other side. if you multiply one side by 4, you have to multiply the other side by 4. if you subtract 3 from one side, you subtract 3 from the other side.<\/p>\n<p>the goal in balancing the equation is isolation. that\u2019s right \u2014 you want to isolate x, or get x (or whatever variable happens to be in the equation) by itself. think of the variable as the unpopular person in the room. everyone \u2013 that is, the other numbers \u2013 want to get away!<\/p>\n<p><span style=\"color: #26b67c;\">\\(4x + 3 = 15.\\)<\/span><\/p>\n<p>to get x by itself, first subtract 3 from both sides of the equation.<\/p>\n<p><span style=\"color: #26b67c;\">\\(4x + 3 &#8211; 3 = 15 &#8211; 3\\)<\/span><\/p>\n<p><span style=\"color: #26b67c;\">\\(4x = 12\\)<\/span><\/p>\n<p>now divide both sides of the equation by 4:<\/p>\n<p><span style=\"color: #26b67c;\">\\(frac {4x} {4} = frac {12} {4}\\)<\/span><\/p>\n<p><span style=\"color: #26b67c;\">\\(x = 3\\)<\/span><\/p>\n<p>let\u2019s try one more:<\/p>\n<p><span style=\"color: #26b67c;\">\\(sqrt{x} = 9\\)<\/span><\/p>\n<p>notice how x has a square root sign on top of it. we want x to be all by itself. therefore we have to get rid of the square root sign by squaring both sides:<\/p>\n<p><span style=\"color: #26b67c;\">\\((sqrt{x})^2 = 9^2 \\)<\/span><\/p>\n<p><span style=\"color: #26b67c;\">\\(x = 81\\)<\/span><\/p>\n<p>the gist? by knowing how to isolate variables by balancing equations, you&#8217;re well on your way to success on the heart of algebra section!<\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h3>inequalities<\/h3>\n<p>although inequalities may seem tricky, if you&#8217;re comfortable balancing equations, you shouldn&#8217;t have much problem with balancing and solving inequalities. <strong>the main way that inequalities differ from equations is that the sign changes if you multiply or divide by a negative number.<\/strong><\/p>\n<p>inequalities come up a lot on sat heart of algebra, so if you need a refresher, make sure to check out this video:<\/p>\n<p><iframe src=\"https:\/\/www.youtube.com\/embed\/h_udngmxk7g\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h3>system of equations<\/h3>\n<p>now, let&#8217;s apply a more advanced version of balancing equations to a very important concept in heart of algebra: system of equations.<\/p>\n<p>system of equations refer to when you have two equations with two unknowns and by combining these two equations, you can solve for these unknowns. there are two methods for combining equations: <strong>substitution and elimination.<\/strong><\/p>\n<p>for a refresher on these methods, check out the below video and then try your hand at solving the practice problem that comes after:<\/p>\n<p><iframe src=\"https:\/\/www.youtube.com\/embed\/urie3bufenm\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h3>algebra basics practice: system of equations<\/h3>\n<p>now, let&#8217;s go back to that original question we started this post off with:<\/p>\n<div class=\"mox\">\n<p>five lemonades and two cookies cost $1.50. two lemonades and five cookies cost $2.70. how much do one cookie and one lemonade cost?<\/p>\n<p><input name=\"choice-q0\" type=\"radio\" value=\"a\" \/>a. $0.10<br \/>\n<input name=\"choice-q0\" type=\"radio\" value=\"b\" \/>b. $0.50<br \/>\n<input name=\"choice-q0\" type=\"radio\" value=\"c\" \/>c. $0.60<br \/>\n<input name=\"choice-q0\" type=\"radio\" value=\"d\" \/>d. $1.50<\/p>\n<details>\n<summary><span style=\"color: #005bc2;\">try to solve the problem yourself before clicking here for the answer!<\/span><\/summary>\n<p>answer: c. $0.60<\/p>\n<p><a href=\"https:\/\/sat.www.catharsisit.com\/questions\/7360\/a\/387393187\">click here for the answer explanation<\/a>!<\/p>\n<\/details>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>coordinate geometry basics<\/h2>\n<p>ok, let\u2019s break down each of the concepts you\u2019ll need to understand, step-by-step, to fully grasp coordinate geometry.<\/p>\n<p>&nbsp;<\/p>\n<h3>quadrants<\/h3>\n<p>there are 4 quadrants in a coordinate plane.<\/p>\n<p>simple, right?<\/p>\n<p>what\u2019s wonky is how those quadrants are arranged on the plane:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-15744\" src=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-1-300x256.png\" alt=\"\" width=\"300\" height=\"256\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-1-300x256.png 300w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-1.png 429w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>a good way to visualize the quadrant layout is to imagine yourself on a cross-country trip in which you start off in new york (the northeast = quadrant i), drive on over to seattle (the northwest = quadrant ii), then drive down to los angeles (southwest = quadrant iii) and finally end up in miami (southeast = quadrant iv).<\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h3>the x-axis &amp; y-axis<\/h3>\n<p>the next things to be familiar with are the <strong>x-axis<\/strong> and the <strong>y-axis<\/strong>. the x-axis runs from left to right (this is the <strong>horizontal<\/strong> line), whereas the y-axis runs up and down (this is the <strong>vertical<\/strong> line).<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-15745\" src=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-2-300x298.png\" alt=\"\" width=\"300\" height=\"298\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-2-300x298.png 300w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-2-150x150.png 150w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-2.png 379w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3>integers and scale of graphs<\/h3>\n<p>next, you need to understand <strong>integers<\/strong>, or what we like to call &#8220;people numbers.&#8221;<\/p>\n<p>why people numbers? think of it this way: you can have one more person or one person fewer. however, you can\u2019t have .3 people or \u00bd a person. in \u201ccoordinate geometry land,\u201d you\u2019ll have integers running from left to right, like on a number line, from negative to positive. on the left hand side the negative numbers will decrease until they hit the middle of the graph (the intersection of the x- and y-axes\u2014or what\u2019s known as <strong>the origin<\/strong>). at that point, you\u2019ll see the number zero. then the numbers continue up from zero.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-15746\" src=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-3-300x298.png\" alt=\"\" width=\"300\" height=\"298\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-3-300x298.png 300w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-3-150x150.png 150w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-3.png 379w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>for the y-axis, the vertical one, the negative numbers start at the bottom, decreasing until they get to zero. then, from the origin going up, each number will go up as well.<\/p>\n<p>on the sat, the scale of the x- and y-axes will vary from question to question. not all axes increase and decrease on an interval of 1. sometimes, the scale of the x-axis will be totally different from the scale of the y-axis! <strong>the test-makers will try to trick you on this point with the answer choices, so definitely make sure to pay attention to the scale of each axis.<\/strong><\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h3>coordinates<\/h3>\n<p>next up are the <strong>coordinates<\/strong>. coordinates are always presented as <strong>two numbers<\/strong>: the first number corresponding to the x-axis, and the second one to the y-axis. for instance, in the graph below, point a has the coordinates (2, 4). that is you \u201cgo over\u201d two points to the right on the x-axis and up four points on the y-axis. point b has the coordinates (-1, -3).<\/p>\n<p>even though the numbers on the x-axis and y-axis are always broken down via integers, coordinates can have fractions, as point c shows below (1\/2, -7\/2).<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-15747\" src=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-4-300x298.png\" alt=\"\" width=\"300\" height=\"298\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-4-300x298.png 300w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-4-150x150.png 150w, \/\/www.catharsisit.com\/hs\/files\/2020\/08\/hoa-4.png 379w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3>slope<\/h3>\n<p>ok here\u2019s the fun part that involves putting together the basics from above.<\/p>\n<p>remember the term \u201crise over run\u201d? you\u2019ve probably had this hammered in for so long you can\u2019t even remember when you first heard it. sadly though, most people forget exactly what it means.<\/p>\n<p>so here it is: you can figure out the slope of a line by seeing how many squares on the coordinate plane one point is above the other. in the graph below, point d has a y-coordinate of 2 and point e has a y-coordinate of -3. so point d is <strong>five squares higher<\/strong> than point e. this is the \u201crise,\u201d how far up it goes from the lowest point, is 5.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-15792\" src=\"\/\/www.catharsisit.com\/hs\/files\/2020\/09\/coordgeom.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2020\/09\/coordgeom.jpg 600w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/coordgeom-300x300.jpg 300w, \/\/www.catharsisit.com\/hs\/files\/2020\/09\/coordgeom-150x150.jpg 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>when you are figuring out the <strong>slope<\/strong>, which consists of a <strong>fraction<\/strong>, always put the \u201crise\u201d in the <strong>numerator<\/strong>.<\/p>\n<p>the \u201crun,\u201d on the other hand, is how far away two points are in a left-to-right sense.<\/p>\n<p>if you count how many squares point d (3, 2) is from point e (-2, -3), in a left to right sense, you\u2019ll get the \u201crun,\u201d which in this case is 5.<\/p>\n<p>maybe you\u2019re wondering why on earth it\u2019s called the \u201crun.\u201d it\u2019s just one of those conventions you have to learn. the key is you always put the \u201crun\u201d, or whatever you want to call it, in the <strong>denominator<\/strong>.<\/p>\n<p>now, the good news. there\u2019s actually a faster way of doing this besides just counting squares! it&#8217;s all in this formula (which your algebra teacher would hate me for leaving out):<\/p>\n<p><center><span style=\"color: #26b67c;\">\\( {y_2 \u2013 y_1} \\)<\/span><br \/>\n<span style=\"color: #26b67c;\">_______<\/span><br \/>\n<span style=\"color: #26b67c;\">\\({x_2 \u2013 x_1}\\)<\/span><\/center>okay, this formula may not seem like the fastest way to get to the slope, but bear with me. students usually get flustered trying to remember which one is the <span style=\"color: #26b67c;\">\\(y_2\\)<\/span> and which one the <span style=\"color: #26b67c;\">\\(y_1\\)<\/span>. but it actually doesn\u2019t matter\u2014<strong>as long as you make sure to start with the same point for both coordinates<\/strong>.<\/p>\n<p>for example, let&#8217;s have <span style=\"color: #26b67c;\">\\(y_2 = 2\\)<\/span> (the y-coordinate in point d) and <span style=\"color: #26b67c;\">\\(y_1 = -3\\)<\/span> (the y-coordinate in point e). that means <span style=\"color: #26b67c;\">\\(x_2 = 3\\)<\/span> (the x-coordinate in point d) and <span style=\"color: #26b67c;\">\\(x_1 = -2\\)<\/span> (the x-coordinate in point e). using the formula, we&#8217;ll get:<\/p>\n<p><span style=\"color: #26b67c;\">\\( 2 \u2013 (-3) \\)<\/span><br \/>\n<span style=\"color: #26b67c;\">_______<\/span><br \/>\n<span style=\"color: #26b67c;\">\\(3 \u2013 (-2)\\)<\/span><\/p>\n<p>which equals:<\/p>\n<p><span style=\"color: #26b67c;\">\\( 5\/5 \\)<\/span> or <span style=\"color: #26b67c;\">\\( 1 \\)<\/span>, meaning that the <strong>slope is 1<\/strong>.<\/p>\n<p>if you switch the order, the equation will look like<\/p>\n<p><span style=\"color: #26b67c;\">\\( -3 &#8211; 2\\)<\/span><br \/>\n<span style=\"color: #26b67c;\">_______<\/span><br \/>\n<span style=\"color: #26b67c;\">\\(-2 &#8211; 3\\)<\/span><\/p>\n<p>which equals:<\/p>\n<p><span style=\"color: #26b67c;\">\\( -5\/-5 \\)<\/span>, which is also <span style=\"color: #26b67c;\">\\( 1 \\)<\/span>.<\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h3>coordinate geometry practice<\/h3>\n<p>every now and then, the sat gets diabolical and throws a coordinate geometry question without the coordinate plane. to make matters worse, the question is usually difficult to begin with\u2014even if you had the plane.<\/p>\n<p><strong>the key is to translate the information by drawing out a mini-coordinate plane. <\/strong>remember, this is a timed test, so it doesn\u2019t have to be anything fancy (don\u2019t worry, the test isn\u2019t going to require you to carefully graph out parabolas the way your math teacher does). as long as your graph can help you better visualize the problem, you\u2019re halfway to getting the problem right!<\/p>\n<p>of course, the other half is always a bit tricky. case in point: the question below. if you&#8217;re able to crack it in less than two minutes, you&#8217;re in <strong>excellent<\/strong> shape for heart of algebra!<\/p>\n<div class=\"mox\">\n<p>1.the center of circle q is the origin of a coordinate plane. if the point (-8, 15) is on the circle, what is the area of an isosceles right triangle with all three of its vertices on the circle?<\/p>\n<p><input name=\"choice-q1\" type=\"radio\" value=\"a\" \/>a. 17<br \/>\n<input name=\"choice-q1\" type=\"radio\" value=\"b\" \/>b. 289\u03c0<br \/>\n<input name=\"choice-q1\" type=\"radio\" value=\"c\" \/>c. 144<br \/>\n<input name=\"choice-q1\" type=\"radio\" value=\"d\" \/>d. 289<\/p>\n<details>\n<summary><span style=\"color: #005bc2;\">click here for the answer and explanation<\/span><\/summary>\n<p>answer: d. 289<\/p>\n<p>view the below video explanation to understand the answer:<\/p>\n<p><iframe src=\"https:\/\/www.youtube.com\/embed\/9v24gwvdpic\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/details>\n<\/div>\n<p>&nbsp;<\/p>\n<p>if you have a comment or question about this problem, leave it for us below!<\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h2>functions<\/h2>\n<p>you\u2019re going to find about 2-6 <a href=\"\/\/www.catharsisit.com\/hs\/sat\/new-sat-math-linear-functions\/\">function<\/a> questions on your sat, and if you haven\u2019t been working with them recently in math class, they might throw you for a loop.<\/p>\n<p>just to be clear, we\u2019re talking about equations that look a bit like this: <span style=\"color: #26b67c;\">\\(g(x) = frac {x-4} 2\\)<\/span><\/p>\n<p>you may have a couple of immediate reactions to equations like this.<\/p>\n<p>for one, you may have the instinct to graph it out right off the bat, which is great, but not necessarily what you\u2019ll need to do on the sat.<\/p>\n<p>or, if you\u2019re not so familiar with functions, you might make a lethal error and assume the <span style=\"color: #26b67c;\">\\(g\\)<\/span> here is a variable. it\u2019s not, by any means, but it&#8217;s also important to note that these functions aren\u2019t always a separate skill from solving algebraic equations. they\u2019re just wearing costumes: <span style=\"color: #26b67c;\">\\(g(x) = frac {x-4} 2\\)<\/span> is the same thing as <span style=\"color: #26b67c;\">\\(y = frac {x-4} 2\\)<\/span>.<\/p>\n<p>although function questions tend to come later on in the section, which is typically where the hardest questions on the test are found, functions are really not that difficult and can be solved pretty quickly with some practice. if it helps, you can change the function notation to look like an algebraic equation that you&#8217;re more familiar with.<\/p>\n<p>try that concept out with relatively simple function question, based on the equation above:<\/p>\n<div class=\"mox\">if <span style=\"color: #26b67c;\">\\(g(x) = frac {x-4} 2\\)<\/span>, and. <span style=\"color: #26b67c;\">\\(g(a) = frac {a} 3\\)<\/span>, what is the value of a?<\/p>\n<details>\n<summary><span style=\"color: #005bc2;\">click here for the answer and explanation<\/span><\/summary>\n<p>answer: <span style=\"color: #26b67c;\">\\(a = 12\\)<\/span><\/p>\n<p>explanation:<\/p>\n<p><strong>if you&#8217;re familiar with the function notation, you can solve the equation as follows: <\/strong><\/p>\n<p><span style=\"color: #26b67c;\">\\(g(a) = frac {a-4} 2 = frac {a} 3\\)<\/span><\/p>\n<p>you can cross multiply to get the following: <span style=\"color: #26b67c;\">\\(2a = 3(a-4)\\)<\/span> or <span style=\"color: #26b67c;\">\\(2a = 3a-12\\)<\/span><\/p>\n<p>isolate <span style=\"color: #26b67c;\">\\(a\\)<\/span> you&#8217;ll get <span style=\"color: #26b67c;\">\\(a = 12\\)<\/span><\/p>\n<p><strong>if you&#8217;re not too familiar with the function notation, it might help to replace <span style=\"color: #26b67c;\">\\(g(x)\\)<\/span> with a <span style=\"color: #26b67c;\">\\(y\\)<\/span>: <\/strong><\/p>\n<p>if <span style=\"color: #26b67c;\">\\(y = frac {x-4} 2\\)<\/span>, at what value of<span style=\"color: #26b67c;\">\\( x\\)<\/span> does <span style=\"color: #26b67c;\">\\(y=x\/3\\)<\/span>?<\/p>\n<p>that&#8217;ll lead you to:<\/p>\n<p><span style=\"color: #26b67c;\">\\(y = frac {x-4} 2 = frac {x} 3\\)<\/span><\/p>\n<p>the algebra from here on will bring you to the same result as it did for the above equation (because they\u2019re the same equation)!<\/p>\n<\/details>\n<\/div>\n<p>&nbsp;<\/p>\n<p>here are some things to keep in mind about functions:<\/p>\n<p>&nbsp;<\/p>\n<h3>the \u201cg\u201d in g(x) is just shorthand<\/h3>\n<p>let\u2019s say you have a friend who has a particular way of eating grapes. let\u2019s call her gina. and let\u2019s say gina brings grapes to school every day, <strong>eats four<\/strong>, splits the rest into <strong>two equal piles<\/strong>, then <strong>throws away one<\/strong> of the piles and <strong>gives away what\u2019s left. <\/strong><\/p>\n<p>the thing is that she brings <strong>a different number<\/strong> of grapes every day. meaning, what she gives away at the end also varies. does this sound familiar? you might recognize this process from above. it\u2019s just an example of that earlier function <span style=\"color: #26b67c;\">\\(g(x) = frac {x-4} 2\\)<\/span>, where:<\/p>\n<ul>\n<li>x is how many grapes gina comes to school with originally<\/li>\n<li><span style=\"color: #26b67c;\">\\(4\\)<\/span> represents how many grapes gina eats<\/li>\n<li><span style=\"color: #26b67c;\">\\(frac {x-4} 2\\)<\/span> represents both the pile that gina throws away and the pile she gives away, and<\/li>\n<li><span style=\"color: #26b67c;\">\\(g(x)\\)<\/span> is just shorthand for gina&#8217;s process: <span style=\"color: #26b67c;\">\\(frac {x-4} 2\\)<\/span><\/li>\n<\/ul>\n<p>what that <span style=\"color: #26b67c;\">\\(g\\)<\/span> really means is a process (not the final result) without writing out each step of the way. gina brings <span style=\"color: #26b67c;\">\\(x\\)<\/span> grapes, and she then \u201c<span style=\"color: #26b67c;\">\\(g\\)<\/span>s\u201d the grapes. in a way, \u201c<span style=\"color: #26b67c;\">\\(g\\)<\/span>\u201d a verb, not a noun.<\/p>\n<p>&nbsp;<\/p>\n<h3>functions inside of functions<\/h3>\n<p>so let\u2019s say you\u2019ve got a whole group of friends with unique grape-eating habits. gina does her grape ritual and then hands off what\u2019s left to hailey. hailey goes through a whole other process. she won\u2019t eat gina&#8217;s grapes unless she gets the same amount of grapes from somebody else, and she then squashes one of them under her foot.<\/p>\n<p>so hailey\u2019s equation would be <span style=\"color: #26b67c;\">\\(h(x) = 2x \u2013 1\\)<\/span>. what she has, then, is described by <span style=\"color: #26b67c;\">\\(h(x)\\)<\/span>. but first we need to think about how many gina gave away.then, put the two processes together into one: <span style=\"color: #26b67c;\">\\(h(g(x))\\)<\/span>. take it from the inside out, and you\u2019ll have no problem.<\/p>\n<p>if hailey&#8217;s grapes are passed along to even <em>more<\/em> friends with specific grape eating habits, just keep nesting the function, e.g. <span style=\"color: #26b67c;\">\\(f(h(g(x))\\)<\/span>. <strong>on the sat, you likely won&#8217;t go beyond one function nested in another. <\/strong><\/p>\n<p>in the end, sat functions like this will end up being <strong>no different<\/strong> than the simpler-seeming sat equations. really.<\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<h3>functions practice: nested practice<\/h3>\n<p>let&#8217;s try out gina and hailey&#8217;s equations with a practice question:<\/p>\n<div class=\"mox\">\n<p>gina brought 20 grapes. if she gives hailey the remainder of her grapes, how many grapes is hailey left with after she goes through her own process?<\/p>\n<details>\n<summary><span style=\"color: #005bc2;\">click here for the answer and explanation<\/span><\/summary>\n<p>answer: 15<\/p>\n<p>explanation:<\/p>\n<p>first, find how many gina will end up giving away with her function, <span style=\"color: #26b67c;\">\\(g(x)\\)<\/span>:<\/p>\n<p><span style=\"color: #26b67c;\">\\(g(20) = frac {20-4} 2 = 8\\)<\/span><\/p>\n<p>then, put that result through the process that hailey uses:<\/p>\n<p><span style=\"color: #26b67c;\">\\(h(g(20)) = h(8) = 2(8) \u2013 1 = 15\\)<\/span><\/p>\n<\/details>\n<\/div>\n<p>&nbsp;<\/p>\n<p><a href=\"#toc\"><img decoding=\"async\" class=\"alignnone size-full wp-image-11102\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/08\/back-to-top-button-1-e1502493700147.png\" alt=\"back to top\" width=\"150\" height=\"30\" \/><\/a><\/p>\n<h2>heart of algebra: you got this!<\/h2>\n<p>the above concepts really do constitute the \u201cheart\u201d of the heart of algebra section, so if you familiarize yourself with them and practice these types of problems, you\u2019ll be in great shape! for more practice problems, make sure to check out <a href=\"\/\/www.catharsisit.com\/hs\/sat\/new-sat-math-algebra-practice-questions\/\">sat math: algebra practice questions<\/a>.<\/p>\n<p>you can also check out the other question types on sat math, like <a href=\"\/\/www.catharsisit.com\/hs\/sat\/sat-math-problem-solving-and-data-analysis\/\">problem solving and data analysis<\/a>, read more about <a href=\"\/\/www.catharsisit.com\/hs\/sat\/sat-math-perfect-800\/\">how to score a perfect 800 on the sat math section<\/a>, and check out some additional <a href=\"\/\/www.catharsisit.com\/hs\/sat\/sat-math-practice\/\">sat math practice<\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>about 33% of questions on sat math will be \u201cheart of algebra\u201d questions. read on for what this means and some practice questions!<\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[91],"tags":[124,81],"ppma_author":[24882],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 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lele","image":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/9f063fdc4c0268d1eac74c41e0906a03","url":"https:\/\/secure.gravatar.com\/avatar\/198545d5a4baea0ba0fd0c0fc8a5d466?s=96&d=mm&r=g","contenturl":"https:\/\/secure.gravatar.com\/avatar\/198545d5a4baea0ba0fd0c0fc8a5d466?s=96&d=mm&r=g","caption":"chris lele"},"description":"chris graduated from ucla with a ba in psychology and has 20 years of experience in the test prep industry. he's been quoted as a subject expert in many publications, including us news, gmac, and business because.","sameas":["https:\/\/www.facebook.com\/chrislelemagoosh1\/","ttps:\/\/www.linkedin.com\/in\/chris-lele-095005a0","https:\/\/twitter.com\/chrismagoosh","https:\/\/www.youtube.com\/channel\/ucz4sgyj6l0cgnvnihy8gy6w"],"knowsabout":["gre","gmat","sat","act","mcat (cars)","lsat"],"jobtitle":"principal curriculum manager","worksfor":"magoosh","url":"\/\/www.catharsisit.com\/hs\/author\/chris\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/h_udngmxk7g","og_video_type":"text\/html","og_video_duration":"836","og_video_width":"480","og_video_height":"360","ya_ovs_adult":"false","ya_ovs_upload_date":"2020-09-11t18:30:51+00:00","ya_ovs_allow_embed":"true"},"authors":[{"term_id":24882,"user_id":10,"is_guest":0,"slug":"chris","display_name":"chris lele","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/198545d5a4baea0ba0fd0c0fc8a5d466?s=96&d=mm&r=g","user_url":"","last_name":"lele","first_name":"chris","description":"chris lele is the principal curriculum manager (and <a href=\"https:\/\/www.amazon.com\/vocabulary-builder-workbook-activities-must-know\/dp\/193975481x\" rel=\"noopener noreferrer\">vocabulary wizard<\/a>) at magoosh. chris graduated from ucla with a ba in psychology and has 20 years of experience in the test prep industry. he's been quoted as a subject expert in many publications, including<a href=\"https:\/\/www.usnews.com\/education\/best-colleges\/slideshows\/10-test-prep-tips-for-sat-and-act-takers\" rel=\"noopener noreferrer\"> us news<\/a>,<a href=\"https:\/\/www.mba.com\/exams-and-exam-prep\/gmat-exam\/why-take-the-gmat-in-2021-when-many-schools-are-test-optional\" rel=\"noopener noreferrer\"> gmac<\/a>, and<a href=\"https:\/\/www.businessbecause.com\/news\/gmat\/3263\/mba-admissions-gre-challenges-gmat\" rel=\"noopener noreferrer\"> business because<\/a>.\r\n\r\n\r\nin his time at magoosh, chris has taught countless students how to tackle the<a href=\"https:\/\/www.amazon.com\/gre-prep-magoosh\/dp\/1939418917\/\" rel=\"noopener noreferrer\"> gre<\/a>, gmat, sat, act, mcat (cars), and lsat exams with confidence. some of his students have even gone on to get near-perfect scores. you can find chris on <a href=\"https:\/\/www.youtube.com\/channel\/ucz4sgyj6l0cgnvnihy8gy6w\">youtube<\/a>,<a href=\"https:\/\/www.linkedin.com\/in\/chris-lele-095005a0\"> linkedin<\/a>, <a href=\"https:\/\/twitter.com\/chrismagoosh\">twitter<\/a> and <a href=\"https:\/\/www.facebook.com\/chrislelemagoosh1\/\">facebook<\/a>!"}],"_links":{"self":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/5483"}],"collection":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/comments?post=5483"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/5483\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=5483"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=5483"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=5483"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=5483"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}