Maths-Notes-Graphing-Techniques.doc

Graphing techniquesRough sketching1. Factorize the equation2. Depending on the power of the factorized equation, sketch the graph a. If the factorized part is to the power of 1, the graph passes through the root (of the equation)b. If the factorized part is to the power of 2, the graph rebounds after touching the rootc. If the factorized part is to the power of 1, the graph has a point of inflection at the rootCirclesThe general equation is x-a2+y-b2=r2 Where r = radius, a,b=center of circleFor the equation of the semicircle (lower and upper half): y= r2-x2EllipsesThe general equation is (x -c)2a2+ (y - d)2b2=1 Where c,d=center of circle Horizontal radius of the ellipse = a Vertical radius of the ellipse = bParabolaThe general equation is y2=cx OR x2=cy OR y=cx2, where c is a constantIn the case of y-a2=cx-b, b,a=turning point, axis of symmetery is y=aIn the case of x-a2=cy-b, a,b=turning point , axis of symmetery is x=aHyperbolaThe general equation is (x -c)2a2- (y - d)2b2=1 OR (y -c)2a2- (x - d)2b2=1The asymptotes are y-d= ba(x-c) OR y-c= ab(x-d)Note that these can be drawn with the CONICS app in the GCRectangular HyperbolaThe general equation is = ax+bcx+d=A+ Bcx+d , where A and B are constantsThe vertical asymptote is x= -dc, while the horizontal asymptote is y=A= acHyperbolic curvesThe general equation is y= ax2+bx+cdx+e=px+q+ rcx+d, where p, q, r are constantsThe vertical asymptote is x= -ed, while the oblique asymptote is y=px+qLinear transformation1. Translationa. Replacing x with x-a results in the graph moving a units in the direction of the positive x-axis (shift right or left)b. Replacing y with y-a results in the graph moving a units in the direction of the positive y-axis (shift up or down)c. Negative translation will occur if a02. Scalinga. Replacing x with xa results in all the x-coordinate of the points in the graph being multiplied by a (Expand or shrink the graph horizontally)b. Replacing y with ya results in all the y-coordinate of the points in the graph being multiplied by a (Expand or shrink the graph horizontally too)3. Reflectiona. Replacing x with -x results in all the x-coordinate of the points in the graph being multiplied by -1 (Reflecting the graph about the y-axis)b. Replacing y with -y results in all the y-coordinate of the points in the graph being multiplied by -1 (Reflecting the graph about the x-axis)When doing transformations, remember to do translation first for changes made to x, and last for changes made to yNon-linear Transformation1. Modulusa. When is f(x) replaced by f(x) , there is a reflection of the parts of the graph below the x-axis only.b. When is f(x) replaced by (x) , there is a reflection of the parts of the graph behind the y-axis only. (The graph is symmetrical about the y-axis)2. Reciprocala. When f(x) replaced by 1f(x) , there are various changes to the graphi. The points where the graph cuts the x-axis become the vertical asymptotes ii. The vertical asymptotes become the points where the graph cuts the x-axisiii. The minimum point becomes the maximum point and vice-versa. iv. The y-coordinate of all the points (and asymptotes) on the graph become their reciprocal. I.e. small points (less than 1) become big, big points become smallv. The x-coordinate of all the points on the graph remains unchangedvi. The original and reciprocal graphs intersect at y= 13. Square roota. When f(x) replaced by f(x) , there are various changes to the graphi. All the points below the x-axis become undefined and are removedii. There are no changes to the roots of the graphiii. The y-coordinate of all the points (and asymptotes) on the graph gets rooted. I.e. small points (less than 1) become biger (still less than 1), big points become smalleriv. The x-coordinate of all the points on the graph remains unchangedv. The original and rooted graphs intersect at y= 1vi. The rooted graph is greater than the original when 0fx14. y2 a. When y is replaced by y2, there are various changes to the graph. (It is equivalent to f(x) )i. The changes are similar to that of the rooted graph (see above)ii. After rooting the graph (as shown above), reflect the rooted graph about the x-axisiii. At the x-intercepts, there are special rules to be applied1) If the original graph just passes through the root, the rooted graph needs to be vertical at the root (i.e. have an “r” shape)2) If the original graph is re