Graphs (Part 2)
Clement Radcliffe,Contributor

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.

REMINDERS

• A quadratic equation is represented graphically by a curve.
• A curve should be drawn by free-hand 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) therefore, f(x) = g(x)

If f(x) = x² - 3x + 2 and g(x) = 2x - 1 then at the point of intersection of the curve and the line, f(x) = g(x).

x² - 3x + 2 = 2x - 1

x² - 3x - 2x + 2 + 1 = 0

x² - 5x + 3 = 0

The x coordinates of the points of intersection are, therefore, the solution of the equation

x² - 3x + 2 = 2x - 1 OR x² - 5x + 3 = 0.

EXAMPLE

Using an appropriate scale, please plot the curve y = 3x2 - 2x - 1. Hence, solve the equations:

a) 3x² - 2x - 1 = 0

b) 3x² - 2x - 1 = 2 - 2x

c) 3x² - 3 = 0 or x² - 1 = 0

SOLUTION

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

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 - 3 = 0, then the equation must be reorganised to the form with 3x² - 2x - 1 on the left-hand side.

a) Given the curve y = 3x2² - 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 = 2 - 2x, then the points of intersection of the curve and the line will represent the solution of the equation 3x² - 2x - 1 = 2 - 2x. From the graph, the solution is x = - 1, 1.

c) Given the equation 3x² - 3 = 0, if the curve y = 3x² - 2x - 1 must be used, then 3x² - 3 = 0 is reorganised as follows:

3x² - 2x + 2x - 2 - 1 = 0.

3x² - 2x - 1 = 2 - 2x

3x² - 2x - 1 + 2x -2 = 0

3x² - 2x - 1 = 2 - 2x

The solution of the equation 3x² - 3 = 0 is the x coordinates of the points of intersection of the curve

y = 3x² - 2x - 1 and the line

y = 2 - 2x.

As in b, x = - 1, 1.

Let us attempt another example.

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

By reorganising the equation 2x² - 2x - 5 = 0, it follows that:

2x² -x -x - 3 -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.

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

From the graph, the turning point is (1/3, -4/3)

The minimum value is -4/3 and is at the point x = 1/3.

A similar approach is used to find the maximum value of F (x) = -2x² +5x+ 3

Please continue to practise, using exercises from your texts.

Enjoy the rest of the week.

Clement Radcliffe is an independent contributor. Send questions and comments to kerry-ann.hepburn@gleanerjm.com