GCSE Revision 101 Maths Solving Quadratics Graphically Daniel

GCSE Revision 101 Maths Solving Quadratics Graphically © Daniel Holloway

The Basics Any quadratic graph has x 2 in its equation We can work out how to plot a quadratic graph using an x and y values table

y = x 2 If we draw out the x and y table for the quadratic equation y = x 2 we should get something like this: x -5 -4 -3 -2 -1 0 1 2 3 4 x 2 25 16 9 4 1 0 1 4 9 16 25 y Next, we plot the points… 35 30 We know that when x = -5, y = 25 25 We know that when x = -4, y = 16 20 And so on for the other values 15 Finally, we connect the points with a smooth curve 10 5 -5 0 5 5 x

Example of Another Curve There may be more complicated graphs to plot involving quadratics Take the graph for y = x 2 + 5 x + 6 x -5 -4 -3 -2 -1 0 1 2 3 x 2 25 16 9 4 1 0 1 4 9 5 x -25 -20 -15 -10 -5 0 5 10 15 6 6 6 6 6 y 6 2 0 0 2 6 12 20 30

Example of Another Curve x -5 -4 -3 -2 -1 0 1 2 3 y 6 2 0 0 2 6 12 20 30 y 35 30 25 20 15 10 5 -5 0 5 x

Roots of Quadratics We can use graphs with quadratics in them to solve quadratic equations When we draw quadratic lines on a graph, it crosses the x-axis at two points. Since the xaxis is the line y = 0, any point along in has a y value of zero We call the “answers” to the equation its roots

Roots of Quadratics Take the graph for y = 2 x 2 - 5 x - 3 for -2 ≤ x ≤ 4 x -2 -1. 5 -1 -0. 5 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 x 2 4 2. 25 1 0. 25 0 0. 25 1 2. 25 4 6. 25 9 12. 25 16 2 x 2 8 4. 5 2 0. 5 0 0. 5 2 4. 5 8 12. 5 18 24. 5 32 5 x -10 -7. 5 -5 -2. 5 0 2. 5 5 7. 5 10 12. 5 15 17. 5 20 -3 -3 -3 -3 y 15 9 4 0 -3 -5 -6 -6 -5 -3 0 4 9 We could plot it and then look at the points at which the line crosses the x-axis

Roots of Quadratics x -2 -1. 5 -1 -0. 5 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 y 15 9 4 0 -3 -5 -6 -6 -5 -3 0 4 9 y With the graph complete, we can easily spot the two points where it crosses the x-axis (although with this graph you could tell these points by looking at the table, usually you will need to draw the graph as they are not integers) 15 10 5 -2 -1 O -5 -10 1 The points are x = -0. 5 and x = 3. 5 x where y 3 = 0. So 2 4 we have solved the equation 0 = 2 x 2 – 5 x – 3 which is 2 x 2 – 5 x – 3 = 0

Square-Root Graphs Because squaring a negative number gives a positive result, there is only one pair of coordinates on a y = x 2 graph for each x value. However, the coordinates of y = √x come in two pairs: � when x = 1, y = ± 1 giving two coordinates: (1, 1) and (1, 1) � when x = 4, y = ± 2 giving two coordinates: (4, 2) and (4, 2)

Square-Root Graphs y 2 1 -1 O -1 -2 1 2 3 4 5 x We can use those points to plot the graph y = √x § x = 0, y = 0 § x = 1, y = 1 § x = 1, y = -1 § x = 2, y = 2 § x = 2, y = -2

Reciprocal Graphs A reciprocal equation takes the form: a y= x All reciprocal graphs have a similar shape and certain symmetrical properties

Reciprocal Graphs Take the graph for 1 y= x y 5 -4 -2 O 2 4 x x y -0. 8 -4 -0. 25 -1. 25 -0. 6 -3 -0. 33 -1. 67 -0. 4 -2 -0. 5 -2. 5 -0. 2 -1 -1 -5 0. 2 1 1 5 0. 4 2 0. 5 2. 5 0. 6 3 0. 33 1. 67 0. 8 4 0. 25 1. 25 Noteisthere is nohelpful value for This not very as xit = 0 doesn’t becauseshow that is very infinity. much. You of acan see that x increases, thescale graph, soas let’s shorten the of the getsx closer valuestoand theadd x axis to the table -5