# Lesson Notes By Weeks and Term - Senior Secondary School 3

STRAIGHT LINE GRAPHS

SUBJECT: MATHEMATICS

CLASS:  SS 3

DATE:

TERM: 1st TERM

REFERENCE TEXTS:

• New General Mathematics for SS book 3 by J.B Channon
• Essential Mathematics for SS book 3
• Mathematics Exam Focus
• Waec and Jamb past Questions

WEEK NINE

TOPIC: STRAIGHT LINE GRAPHS

CONTENT

• Gradient of a Straight Line.
• Drawing of Tangents to a Curve.

The gradient (or slope) of a straight line is a measure of the steepness of the line.

The gradient of a line may be positive or negative.

Consider line LM shown in the diagram below. The line slopes upwardsto the right and it makes an acute angle of with the positive x-axis, so tan is positive. The gradient of the line can be found by choosing any two convenient points such as A and B on the line. In moving from A to B, x increases () and y also incrases ().

i.e. increase in x = horizontal distance = AC

increase in y = vertical distance = BC

the gradient of a line is represented by letter m.

the gradient of a line LM is given by:

m = increase in y from A to B increase in x from A to B= BCAC= 24= 12

also in ABC, tan = BCAC= 24= 12

it follows that the gradient of line AB = tan . When a line slopes upwards (uphill) to the right, the gradient of the line is positive.

In the diagram below, line PQ slopes downwards and it makes an obtuse angle with positive x-axis, so tan is negative. Again, to find the gradient of the line, we choose two convenient points such as D and F on the line. In moving from D to F, x increase () and y decreases ().

i.e. increase in x = horizontal distance = DE and decrease in y = vertical distance = EF.

The gradient, m of line PQ is given by:

m = increase in y from D to Fincrease in x from D to F= EFDE= -34= -34

Also the gradient of line PQ = tan

When a line slopes downwards to the right (i.e. downhill) the gradient is negative.

For example, in the diagram below, the slope goes up 3 units for every 4 units across. Since triangles PQT, QRU and RSW are similar,

We have: QTPT= RUQU= SWRW= 34

This means the gradient of the line is given by:

m=QTPT or m= RUQU or m= SWRW

Where the letter ‘m’ represents gradient.

Calculating the Gradient of a Line

The gradient of a straight line can be calculated from any two points on the line.

Let the two points on line PQ be A and B. if the coordinates of point A are (x1, y 1) and the coordinates

of point B are (x2, y2), then in moving from A to B, the increase in x (or change in x) is AC and the increase in y (or change in y) is CB, i.e. AC = x2 – x1 and CB = y2 – y1,

Thus, the gradient, m of the line PQ is given by:

m = increase in y increase in x = difference in y coordinate difference in x coordinate

= CBAC= y2-y1x2-x1

Exercise

Calculate the gradient of the line joining the points C(-2, -6) and D(3, 2) and.

Solution

Method 1

Plot the points C(-2, -6) and D(3, 2).

Draw a straight line to pass through the points.

Gradient = increase in yincrease in x= EDCE= 85

Method 2

We can calculate the gradient in the following 2 ways.

1. In moving from C to D

(x1, y1) = (-2, -6) and (x2, y2) = (3, 2)

m = y2-y1x2-x1= 2-(-6)3-(-2)= 2+63+2= 85

1. In moving from D to C

(x1, y1) = (3, 2) and (x2, y2) = (-2, -6)

m = y2-y1x2-x1= -6-2-2-3= -8-5= 85

Notice that the answer is the same in obht cases, therefore, it does not matter which point we call the first or the second.

Example

Find the gradient of the line joining (-4, 6) and (3, 0)

Solution

(x1, y1) = (-4, 6) and (x2, y2) = (3 , 0)

m = y2-y1x2-x1= 0-63-(-4)= -63+4= -67= -67

Evaluation

Find the gradients of the line joining the following pairs of points.

1.   (9,7) , (2,5)
2.   (2,5) , (4,5)
3.   (2,3) , (6,-5)

Drawing the Graphs of Straight Lines

Example

(a) Draw the graph of 3x + 2y = 8

(b) Find the gradient of the line.

Solution

(a) First make y the subject.

3x + 2y = 8

2y = 8 – 3x

y = 8-3x2

Choose three easy values and then make a table of values as shown below.

When x = 0,    y = 8-02=4

When x = 2,       y = 8-62= 22=1

When x = 4,    y =8-122= -42= -2

 x 0 2 4 y 4 1 -2

The graph of 3x + 2y = 8 is shown below.

(b) Choose two easy points such as P and Q on the line.

Gradient of PQ = increase in yincrease in x= -RQPR

-64= -32

Evaluation

Using three convenient points, draw the graph of the following linear equations and then find their gradients.

1. 2x-y-6=0       2.) 5y+4x=20    3.)  3x-2y=9

The diagram above represents the graph of the function y =2x2 + x – 5.

The gradients at P and S can be found as follows:

Gradient at P = gradient of tangent PQ. By constructing a suitable right-angled triangle with hypotenuse PQ, the gradient is Gradient = PRPQ= -71= -7

Remember that the gradient is negative because the tangent slopes downwards from left to right.

By constructing a suitable right-angled triangle with hypotenuse ST, the gradient is

Remember that the gradient is positive because the tangent slopes upwards from left to right.

Evaluation

Draw the graphs of the following functions and use the graphs to find the gradients at indicated points.

1) y= x2 –x-2   at   x= -1

2) y= x2-3x-4=0   at x = 4

GENERAL EVALUATION/ REVISION QUESTIONS

1. A straight line passes through the points (3,k) and (-3,2k). If the gradient of the line is -2/3, find the value of k. What is the equation line?
2. Sketch the following graphs using gradient-intercept method.
3. a) y= 0.5x - 3      b) y= 5x       c)  y = x/4   -  3      d)   2y-10  = 2x
4. Find the gradients of the curves at the points indicated.
5. a) y= 6x -  x2   at  x= 3       b)   x2 – 6x + 5

WEEKEND ASSIGNMENT

1. Find the gradient of the equation of line 2y – 10 = 2x   A. 1   B.  2    C.  3   D.    4
2. Find the gradient of the line joining (7,-2) and (-1,2)    A. ½   B. – ½   C.  1/3  D. -1/3
3. Find the equation of a straight line passing through (-3,-5) with gradient 2.
4. y =3x-1  B. y=2x-1C. y=2x-1   D. y=3x+1

Given that 3y-6x +15=0, use the information to answer questions 4 and 5.

1. Find the gradient of the line.     A. 5  B.  -5     C.   2    D.  -2
2. Find the intercept of the line. A. 5  B.  -5     C.   2    D.  -2

THEORY

1. Draw the graph of y= 2x-3 using convenient points and scale. Hence , find the gradient of the line at any convenient point.

2a) Copy and complete the following table of values for the relation y= 2x2 – 7x-3.

 X -2 1- 0 1 2 3 4 5 Y 19 -3 -9
1. b) Using 2cm to 1unit on the x-axis and 2cm to 5units on the y-axis, draw the graph of y= 2x2 -7x-3 for -2x≤5.
2. c) From your graph, find the:
3. minimum value of y.
4. the equation of the line of symmetry.

iii. the gradient of the curve at x=1.