1
FUNDAMENTAL MATHEMATICS
Study Guide 3
GRAPHS
Prepared by Dr. Faleye Sunday
, 2
ii
MODULE 3: GRAPHS
Unit 1 Analytic Geometry 1
Outcomes 1
1.1 Graphs 3
1.2 Cartesian Coordinates and Graphs in the Cartesian Plane 14
1.3 Formules we often use 34
Unit Summary 56
Checklist 57
Unit 2 Relations and Functions 59
Outcomes 59
2.1 Relations and functions in R x R 60
2.2 Combining Functions 86
Unit Summary 94
Checklist 96
Unit 3 Straigtht line Graphs 97
Outcomes 97
3.1 Drawing Lines 98
3.2 Finding Equations of Lines 125
3.3 Applications of Lines and Linear Functions 136
Unit Summary 167
Checklist 171
, 3
iii MAT0511/001
Unit 4 Parabolas 173
Outcomes 173
4.1 Characteristics of Parabolas defined by y = a(x — h)2+k 174
4.2 Sketching Parabolas 197
4.3 Finding the Equation of a Parabola 214
4.4 Using Parabolas and Quadratic Functions 222
Unit Summary 231
Checklist 233
Unit 5 Hyperbolas 235
Outcomes 235
5.1 Characteristics of Hyperbolas 235
5.2 Inverse Proportion 253
Unit Summary 257
Checklist 258
Unit 6 Combinations of Graphs 260
Outcomes 260
6.1 Graphs, Graphs and More Graphs 260
Unit Summary 278
Checklist 280
ANSWERS 281
REFERENCES 304
, 4
1 MAT0511/001
UNIT 1: The Set of Real Number
OUTCOMES
After studying this unit you should be able to do the following.
1.1 : Graphs
► Choose an appropriate scale (if an accurate graph is required) or mark vertical and horizontal lines in
a suitable way (if an illustration of information is required) and plot given data, taking into account
whether the graph consists of separate dots, or whether the dots can be joined in some way.
► Interpret information conveyed by a given graph.
1.2 : Cartesian Coordinates and Graphs in the Cartesian Plane
► Represent solutions of equations or inequalities in one variable by means of a one–dimensional graph
(i.e. on a number line).
► Find coordinates of a given point in the Cartesian plane, or, if the coordinates are known, locate the
point.
► Use a table of values to draw a graph.
► Use an equation to set up a table of values and hence draw a graph.
► If a graph represents an equation
D find the x – and y–intercepts of the graph
D determine whether or not a given point lies on the graph, i.e. whether or not the coordinates of the
point satisfy the equation of the graph
D find values of x for which y > 0, y = 0 (i.e. the x –intercept) or y < 0
D recognise from the quadrant in which a point lies whether x > 0 or x < 0 and whether y > 0 or
y < 0.
1.3 : Formulas we often use
► Use the Theorem of Pythagoras.
► Use the distance formula.
► Use the midpoint formula.
► Find the standard equation of a circle with radius r and centre at (0, 0) .
FUNDAMENTAL MATHEMATICS
Study Guide 3
GRAPHS
Prepared by Dr. Faleye Sunday
, 2
ii
MODULE 3: GRAPHS
Unit 1 Analytic Geometry 1
Outcomes 1
1.1 Graphs 3
1.2 Cartesian Coordinates and Graphs in the Cartesian Plane 14
1.3 Formules we often use 34
Unit Summary 56
Checklist 57
Unit 2 Relations and Functions 59
Outcomes 59
2.1 Relations and functions in R x R 60
2.2 Combining Functions 86
Unit Summary 94
Checklist 96
Unit 3 Straigtht line Graphs 97
Outcomes 97
3.1 Drawing Lines 98
3.2 Finding Equations of Lines 125
3.3 Applications of Lines and Linear Functions 136
Unit Summary 167
Checklist 171
, 3
iii MAT0511/001
Unit 4 Parabolas 173
Outcomes 173
4.1 Characteristics of Parabolas defined by y = a(x — h)2+k 174
4.2 Sketching Parabolas 197
4.3 Finding the Equation of a Parabola 214
4.4 Using Parabolas and Quadratic Functions 222
Unit Summary 231
Checklist 233
Unit 5 Hyperbolas 235
Outcomes 235
5.1 Characteristics of Hyperbolas 235
5.2 Inverse Proportion 253
Unit Summary 257
Checklist 258
Unit 6 Combinations of Graphs 260
Outcomes 260
6.1 Graphs, Graphs and More Graphs 260
Unit Summary 278
Checklist 280
ANSWERS 281
REFERENCES 304
, 4
1 MAT0511/001
UNIT 1: The Set of Real Number
OUTCOMES
After studying this unit you should be able to do the following.
1.1 : Graphs
► Choose an appropriate scale (if an accurate graph is required) or mark vertical and horizontal lines in
a suitable way (if an illustration of information is required) and plot given data, taking into account
whether the graph consists of separate dots, or whether the dots can be joined in some way.
► Interpret information conveyed by a given graph.
1.2 : Cartesian Coordinates and Graphs in the Cartesian Plane
► Represent solutions of equations or inequalities in one variable by means of a one–dimensional graph
(i.e. on a number line).
► Find coordinates of a given point in the Cartesian plane, or, if the coordinates are known, locate the
point.
► Use a table of values to draw a graph.
► Use an equation to set up a table of values and hence draw a graph.
► If a graph represents an equation
D find the x – and y–intercepts of the graph
D determine whether or not a given point lies on the graph, i.e. whether or not the coordinates of the
point satisfy the equation of the graph
D find values of x for which y > 0, y = 0 (i.e. the x –intercept) or y < 0
D recognise from the quadrant in which a point lies whether x > 0 or x < 0 and whether y > 0 or
y < 0.
1.3 : Formulas we often use
► Use the Theorem of Pythagoras.
► Use the distance formula.
► Use the midpoint formula.
► Find the standard equation of a circle with radius r and centre at (0, 0) .
