Lesson BASICS - Sketching Quadratics
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<b></b> <b>Introduction</b> Quadratics are a family of curves that all have the same basic shape and are always symmetrical: in terms of maths, they are simple to work with. <b>Basic Shape</b> A quadratic is one of two shapes: {{{graph(200,200,-2,2,-5,5,x^2)}}} or {{{graph(200,200,-2,2,-5,5,-x^2)}}} I shall call these "u-shaped" or "n-shaped". <b> a u-shaped quadratic occurs when the {{{x^2}}} term is POSITIVE a n-shaped quadratic occurs when the {{{x^2}}} term is NEGATIVE </b> So, now you know what shape {{{y=-3x^2+5x-8}}} is, instantly... it is n-shaped. <b>Finding info to sketch</b> Just by looking at the equation, we know the shape. What we need to know is where is the curve in relation to the axes... specifically, where (if anywhere) does it cross the x-axis? <b>the point(s) where ANY equation crosses the x-axis is/are the SOLUTIONS/ROOTS</b>. The act of "factorising or using the quadratic formula" to solve quadratics is to find those points where the equation crosses the x-axis, so we can then sketch it. How to factorise is contained within other Lessons. What else do we need to know? Well, we have the shape and the roots...we can SKETCH the curve now. Remember, it is symmetrical, always. By this i mean that the turning point in the curve is ALWAYS mid-way between the roots. Once we have drawn our sketch, we need to add a couple more bits of information, namely: 1. the crossing point on the y-axis --> the y-intercept 2. the x- and y-values of the turning point. <b>y-intercept</b> Crossing the y-axis means x=0, so put x equal to zero into the equation, and you get the y value. <b>Q</b> find the y-intercept of {{{y=3x^2-7x+13}}}. <b>A</b> putting x=0 into this gives y=13, straight away. Easy! <b>Turning Point coordinates</b> Once we have found the roots, say at x=-2 and x=6, we know that the turning point is at the midpoint, namely (-2+6)/2 = 2 so, when x=2, put this value into the equation, to find y. This is all you need to know about sketching Quadratics.
