Maths Solving Quadratics Graphically GCSE Revision 101 Maths Solving Quadratics Graphically © Daniel Holloway

The Basics Any quadratic graph has x2 in its equation We can work out how to plot a quadratic graph using an x and y values table

y = x2 If we draw out the x and y table for the quadratic equation y = x2 we should get something like this: x -5 -4 -3 -2 -1 1 2 3 4 5 x2 25 16 9 y y Next, we plot the points… 35 We know that when x = -5, y = 25 30 We know that when x = -4, y = 16 25 20 And so on for the other values 15 Finally, we connect the points with a smooth curve 10 5 -5 5 x

Example of Another Curve There may be more complicated graphs to plot involving quadratics Take the graph for y = x2 + 5x + 6 x -5 -4 -3 -2 -1 1 2 3 x2 25 16 9 4 5x -25 -20 -15 -10 5 10 15 6 y 12 20 30

Example of Another Curve -5 -4 -3 -2 -1 1 2 3 y 6 12 20 30 y 35 30 25 20 15 10 5 -5 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 x-axis 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 = 2x2 - 5x - 3 for -2 ≤ x ≤ 4 We could plot it and then look at the points at which the line crosses the x-axis x -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 x2 2.25 0.25 6.25 9 12.25 16 2x2 8 4.5 12.5 18 24.5 32 5x -10 -7.5 -5 -2.5 5 7.5 10 15 17.5 20 -3 y -6

Roots of Quadratics x -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 y 15 9 -3 -5 -6 y 15 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) The points are x = -0.5 and x = 3.5 where y = 0. So we have solved the equation 0 = 2x2 – 5x – 3 which is 2x2 – 5x – 3 = 0 10 5 O x -2 -1 1 2 3 4 -5 -10

Square-Root Graphs Because squaring a negative number gives a positive result, there is only one pair of coordinates on a y = x2 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 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 2 1 -1 O x 1 2 3 4 5 -1 -2

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

Reciprocal Graphs 1 Take the graph for y = x x y -0.8 -1.25 -0.6 -1.67 -0.4 -2.5 -0.2 -5 0.2 5 0.4 2.5 0.6 1.67 0.8 1.25 x y -4 -0.25 -3 -0.33 -2 -0.5 -1 1 2 0.5 3 0.33 4 0.25 Take the graph for y = x y 5 -4 -2 O 2 4 x This is not very helpful as it doesn’t show very much of a graph, so let’s shorten the scale of the x values and add to the table Note there is no value for x = 0 because that is infinity. You can see that as x increases, the graph gets closer to the x axis -5