FUNCTIONS

Concepts in Functions

Straight Line Graphs

Parabolas

Hyperbolas

Exponentials

Sine Graphs

Cos Graphs

Tan Graphs

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STRAIGHT LINE GRAPHS

The graph of a linear function is a

straight line.

Standard form of a straight line graph:

y = m x + c

where m = gradient

and c = y-intercept

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Effects of m and c in y = mx + c:

m represents the gradient of the straight line

If m > 0, the gradient is positive

If m < 0, the gradient is negative

c represents the y-intercept of the straight line graph and also indicates the vertical translation of the graph

If c > 0, the graph has a positive y-intercept

and shifts up by c units

If c < 0, the graph has a negative y-intercept

and shifts down by c units

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Sketching Straight Line Graphs Using The

Table Method

Example 1

Sketch the graph of y = 2x+4 by using the table

method.

x -1 0 1 2

y 2 4 6 8

• The x-values were randomly chosen.

• The y-values were found by substituting the x -

values into the equation y = 2x +4.

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Note!

x-intercept is at

(-2; 0)

y-intercept is at

(0; 4)

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Example

Sketch the graph of y = -3x - 3 by using the dual-

intercept method.

x-intercept: y = 0 y-intercept: x = 0

0 = -3x – 3 y = -3(0) – 3

3x = -3 y = 0 – 3

x = -1 y = -3

(-1; 0) (0;-3)

Sketching Straight Lines using the Dual-Intercept Method

Sketching Straight Line Graphs Using The

Dual-Intercept Method

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The Graph of y = -3x – 3:

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Sketching Straight Line Graphs Using The Gradient-

Intercept Method

Example

Sketch the graph of by using the gradient-

intercept method.

xxg2

1)(

• The negative sign

indicates that the line

slopes to the left. The y-

intercept is 0.

• The numerator tells us

to rise up 1 unit from

the y - intercept.

• The denominator tells

us to run 2 units to the left

Determining the Gradient using the Gradient-Intercept Method 8

Horizontal and Vertical Lines

a) Sketch the graph of y = 2

In this relation, the x-values can vary but the

y - values must always remain 2.

Lines which cut the y - axis

and are parallel to the x – axis

have equation y = number

x -1 0 1 2

y 2 2 2 2

Horizontal Straight Lines Determining the Gradient

of Horizontal Straight Lines 9

b) Sketch the graph of x = 2

In this relation, the y-values can vary but the

x - values must always remain 2.

Lines which cut the x - axis

and are parallel to the y – axis

have equation x = number

x 2 2 2 2

y -2 -1 0 1

Vertical Straight Lines

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1. Draw neat sketch graphs of the following

linear functions on separate axes.

(a) f (x) = 3x- 6 (f) y - 3x = 6

(b) g (x) = - 2x + 2 (g) y = 3x + 2

(c) h (x) = - 4x (h) 2x + 3y + 6 = 0

(d) 5x + 2y = 10 (i) 3x + 3y = 0

(e) y – x = 0 (j) 3x = 2y

Straight Line Graphs Calculator

EXERCISE

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2. With each of the equations below, do the

following:

i. Write the equation in standard form

ii. Determine the gradient

iii. Determine the y-intercept

(a) 2y - 4x = 0 (c) 6x - 3y = 1

(b) 2y + 4x = 2 (d) x - 2y = 4

3. Given the following equation: f (x) = - 2x + 1

(a) Sketch the graph

(b) Draw the graph of g(x) if it is f(x) the

has been translated 2 units upwards.

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Example 1: Determine the equation of the line:

The y-intercept is 3, so c = 3.

y = m x + 3

Substitute the point (8 ; - 1)

- l = m(8) + 3

- 1 = 8m + 3

- 8m = 4

m = -½

Therefore the equation is :

y = - ½ x +3

Determining the Equation of a Straight Line Graph

Finding the Equations of Straight Line Graphs

Calculating the Gradients of

Different Staircases Challenge!Finding the gradient when c = 0 13

EXERCISE 1. Determine the equations of the following lines:

(a) (b)

(c ) (d)

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2. Determine the equation of the line:

a) passing through the point (-1; - 2) and cutting the y-axis at 1.

b) Determine the equation of the line with a gradient of - 2 and passing through the point (2; 3).

c) Determine the equation of the line which cuts the x-axis at 5 and the y - axis at - 5.

d) Determine the equation of the line which cuts the x-axis at – 3 and the y - axis at 9.

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Parallel Lines Parallel lines have the same gradient (m)

1. Are these lines parallel? y = 2x + 4 and y – 2x – 6 = 0

• y = 2x + 4 => m = 2

• y – 2x – 6 = 0

y = 2x + 6 => m = 2 Yes! They are parallel.

2. Are these lines parallel? 2y = 2x + 4 and – 2x + y – 6 = 0

• 2y = 2x + 4

y = x + 2 => m = 1

• - 2x + y – 6 = 0

y = 2x + 6 => m = 2 No! They are not parallel.

Parallel lines 16

Perpendicular Lines

Perpendicular Lines 17

Point of Intersection:

The point at which both straight line graphs have the same value for x and for y

Can determine the point of intersection

a) graphically

b) algebraically (solve equations simultaneously)

Example:

Solve 3x-y=4 and 2x-y=5

a) Graphically

Point of intersection

Intersecting Straight Line Graphs

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Example: Solve 3x-y=4 and 2x-y-5

b) Algebraically

3x – y = 4 … (1)

-y = 4 – 3x

y = 3x – 4 … (3)

2x-y=5 … (2)

Substitute (3) into (2)

2x – (3x-4) = 5

2x – 3x + 4 = 5

-x = 1

x = -1

Substitute x = -1 into (3)

y=3(-1)-4

y =-7

Point of intersection: (-1;-7)

Points of Intersection

Calculator 19

EXERCISE

1. Draw neat sketch graphs of the following lines on the same set of axes:

x + y = 3 and x - y = -1.

2. Solve x + y = 3 and x - y = - 1 using the method of simultaneous equations.

3. Determine the coordinates of the point of

intersection of the following pairs of

lines: x + 2y = 5 and x- y = - 1

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