Reminders

THIS WEEK, we will complete the review of algebra by considering aspects of graphs. Specifically, it is my intention to elaborate on the solution of quadratic equations, using a graph. A quadratic equation is represented graphically by a curve. The curve is usually plotted against perpendicular x and y axes. The axes should be labelled and appropriate scale(s) used. A curve should be drawn by freehand sketch. The x axis has the equation y = 0 and the y axis has the equation x = 0. Given the curve y = f(x) and the line y = g(x), then the points of intersection of both are represented by: y = f(x) = g(x) f(x) = g(x) If f(x) = x2 + 2x - 3 and g(x) = 0 (x axis)

Then the points of intersection of the curve and the line is represented by : x2 + 2x - 3 = 0

Therefore, the solution of this equation is the x coordinates of the points of intersection of the curve and the x axis. If f(x) = x2 + 2x - 3 and g(x) = 2x - 2

Then the points of intersection of the curve and the line is also represented by: f(x) = g(x). x2 + 2x - 3 = 2x - 2 x2 + 2x - 2x - 3 + 2 = 0 x2 - 1 = 0 The x coordinates of the points of intersection is, therefore, the solution of the equation x2 + 2x - 3 = 2x - 2 OR x2 - 1 = 0.

The following are the graphs of f(x) and g(x) illustrating the points of intersections. x 0 1 2 y -2 0 2 y = x2 + 2x - 3

Points A and B represent the points of intersection and have coordinates A( 1 , 0 ) and B (-1 , - 4 ). The x coordinates, 1 or - 1 are the solutions. The following is another example for your review.

EXAMPLE

Using an appropriate scale, plot the curve y = 3x2 - 2x - 1. Hence solve the equations: a) 3x2 - 2x - 1 = 0 b) 3x2 - 2x - 1 = 4x - 1

SOLUTION

Given the equation y = 3x2 - 2x - 1, we complete the table:

GRAPH 2

a) Given the curve y = 3x2 - 2x - 1, the solution of the equation 3x2 - 2x - 1 = 0 is the x values of the points of intersection of the curve y = 3x2 - 2x - 1 and the line y = 0 or the x axis. The solution is x= 1, - .33

b) Given the curve y = 3x2 - 2x - 1, by plotting the line y = 4x 1, then the points of intersection of the curve and the line will represent the solution of the equation 3x2 - 2x - 1 = 4x - 1.

Using the same axes, plot the line y = 4x - 1. From the graph, the solution is x = 0 or 2.

POINTS TO NOTE

Given the curve y = 3x2 - 2x - 1, then the curve may be used to solve any equation as long as 3x2 - 2x - 1 is on one side of the equation.

To solve the equation, 3x2 - 2x - 1 = 4x - 1, then the equation must be reorganised as follows: 3x2 - 2x - 1 = 4x - 1. 3x2 - 2x - 1 - 4x + 1 = 0 3x2 - 6x = 0 3x(x - 2) = 0 The solution of 3x2 - 2x - 1 = 4x - 1 and 3x(x - 2) = 0 is the same that is: x = 0 or 2. Let us attempt another example.

EXAMPLE

Given the curve y = 2x2 - x - 3, solve the equation 2x2 - 2x - 5 = 0.

SOLUTION

If you are to use the graph, y = 2x2 - x - 3, then 2x2 - x - 3 must be on one side of the equation. By reorganising the equation 2x2 - 2x - 5 = 0, it follows that: 2x2 -x-x - 3 -2 = 0 2x2 -x- 3 - x - 2 = 0 2x2 -x- 3 =x + 2 Then the solution of 2x2 - 2x - 5 = 0 is the x coordinates of the points of intersection of the curve y = 2x2 - x - 3 and the line y= x + 2.

By plotting the line y= x + 2, the solution of the given equation is found by reading off the points of intersection.

EXAMPLES

i) Given the plot of the graph, y = x2 + 3x + 2 use it to solve the equation, x2 +2x +1 =0 ii) Given the plot: y = 2x2 - 3x + 4 hence solve, 2x2 - 5x + 1 = 0 Let us now review the application of graph to find maximum and minimum values.

Given the function f(x) = 2x2 -x- 3 , the minimum value may be found using the graph y = 2x2 -x- 3. This value is found by the determination of the coordinates of the turning points of the curve. Given the turning point M(x , y), then the x coordinate is the position of the minimum value and the y coordinate is the minimum value of the function f(x).

Plotting the graph y = 2x2 -x- 3

GRAPH 3

From the graph the turning point is (1/4, - 13/4)

The minimum value of f(x) is - 13/4 and is at the point is (1/4, - 13/4).

The axis if symmetry is the vertical line through the turning point. In this case it is x = 1/4. A similar approach is used to find the maximum value of f(x) = - 2x2 +5x+ 3 Please attempt the following example on your own: 1. Given the function f(x) = - 4x2 + 3x + 2, solve: i) - 4x2 + 3x + 2 = 0 ii) - 4x2 - x+ 1 = 0 Please explain your method.