Let y = f(x) be a function of x. For every x in the domain of f, there exists an ordered pair (x, f(x)). The set of points (x, f(x)) is called the graph of f.

By plotting these points on the Cartesian plane, we obtain the graph of f. y = f(x) is called the equation of the graph.

Plot the points (1,√8 ), (2, √5), (3,0) (-1, √8), (-2 ,√5), (-3, 0) and join them to form a smooth curve.

y=f(x) =

The graph is a semi-circle above the x-axis with radius 3 cm.

we get a semi-circle below the x-axis.

1.6.2 Graph of the step function or greatest integer function

If y = f(x) = [x]

then f(x) = [x]

= n for n ≤ x < n + 1

Consider

Consider

f = 0 for 0 ≤ < 1

f = 0 for 0 ≤ < 1

f = 0 for 0 ≤ < 1

Therefore, we find that the ordered pairs are

The graph of the step function is as follows: y = f(x) = [x]

The points (1,1), (2,2), (3,3), (-1,-1), (-2,-2) are included.

The graph of y = f(x) = | x |

x |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |

| x | |
| -3 | |
| -2 | |
| -1 | |
| 0 | |
| 1 | |
| 2 | |
| 3 | |

y |
3 |
2 |
1 |
0 |
1 |
2 |
3 |

Plot the points (-3,3), (-2,2), (-1, 1), (0,0) (1,1), (2,2), (3,3) and join them to get the following figure.

Graph of a^{x} (a>0).

**Case (i)** Let a > 1. For all positive or negative x ∈R, ax is always positive. Consider a = 2.

y = f(x) = 2^{x}

x |
-2 |
-1 |
0 |
1 |
2 |

2 |
2 |
2 |
2 |
2 |
2 |

y |
1/4 |
1/2 |
1 |
2 |
4 |

Plot the points (-2 , 1/4), (-1,1/2), (0,1),(1,2) (2,4) on the graph and join them to form a smooth curve.

The graph of y = ax for any a > 1 is as follows.

**Case (ii)** Let 0 < a < 1 for all positive or negative x ∈R, a^{x} is always positive. The graph can be drawn as:

Consider a = ½, then 0 < 1/2 < 1.

If y = f(x) = (1/2)^{x}

then

x |
-2 |
-1 |
0 |
1 |
2 |

(1/2) |
(1/2) |
(1/2) |
(1/2) |
(1/2) |
(1/2) |

y |
4 |
2 |
1 |
1/2 |
(1/4) |

since (1/2)-^{2} = (2)^{2} = 4

Plot the points (-2,4), (-1,2), (0,1), (1,1/2), (2,1/4) and join them to form a smooth curve.

Draw the graph of y = log_{a}x (a ≠ 0, a > 0)

The graph of y = log_{a} xa > 1

Consider a = 2 then y = f(x) = log_{z}x

x |
1/4 |
1/2 |
0 |
1 |
4 |

log2 |
log |
log |
log |
log2 |
log |

y |
-2 |
-1 |
0 |
1 |
2 |

Plot the points (1/4, -2), (1/2, -1), (0,0), (1,1), (4,2) on the graph.

When 0 < a < 1, the graph of y = loga^{x} is

In particular, let a = 1/2.

y = log1/2x

x |
1/4 |
1/2 |
1 |
2 |
4 |

log1/2 |
log |
log |
log |
log |
log |

y |
2 |
1 |
0 |
-1 |
-2 |

Since 2 = (1/2)-1 and 4 = (1/2)-2,

plot the points (1/4,2), (1/2,1), (1,0), (2,-1), (4,-2) and join them to obtain the graph.

It is preferable if the student studies the graphs of exponential function f(x) = a^{x} and logarithmic functions f(x) = logax with the help of Section 2.

Sketch the graph of

Let y

We cannot take **negative** values for x because no real number exists whose square is negative.

x |
0 |
1 |
4 |
9 |
16 |

√x |
√0 |
√1 |
√4 |
√9 |
√16 |

y |
0 |
±1 |
±2 |
±3 |
±4 |

Plotting the points (0,0), (1,-1), (1,1), (4,-2), (4,2), (9,-3), (9,3) and (16,-4), (16,4) and joining them we get the following graph.

Sketch the graph of y = 1/x.

x |
-5 |
-2 |
-1 |
1 |
2 |
5 |

1/x |
1/-5 |
-1/2 |
-1 |
1 |
1/2 |
1/5 |

y |
-0.2 |
-0.5 |
-1 |
1 |
0.5 |
0.2 |

On plotting the points (-5, -0.2), (-2, -0.5), (-1, -1), (1,1), (2, 0.5), (5, 0.2) and joining them, we get:

Sketch the Graphs of the following functions:

- y =
- y =
- y = 1/2x
- y = 3
^{x} - y = log3
^{x}

- y =√x
^{4}x-3

-2

-1

0

1

2

3

x

^{4}(-3)

^{4}(-2)

^{4}(-1)

^{4}(0)

^{4}(1)

^{4}2

^{4}3

^{4}y

±9

±4

±1

0

±1

±4

±9

Plot the points (-3,9), (-3,-9), (-2,-4), (-2,4), (-1,-1), (-1,1), (0,0), (1,1), (1,-1), (2,-4), (3,9) and join them.

- y =

Taking only the positive square roots.

x-2

-1

0

1

2

x

^{2}(-2)

^{2}(-1)

^{2}(0)

^{2}(1)

^{2}(2)

^{2}4-x

^{2}4-4

4-1

4-0

4-1

4-4

y

0

1.7

2

1.7

0

Joining the points (-2,0), (-1,1.7), (0,2), (1,1.7), (2,0) we get the following graph.

- y = 1/2x

x-3

-2

-1

1

2

3

2x

-6

-4

-2

2

4

6

1/2x

-1/6

-1/4

-1/2

1/2

1/4

1/6

y

-0.16

-0.25

-0.5

0.5

0.25

0.16

Plotting the points (-3,-0.16), (-2,-0.25), (-1,-0.5), (1,0.5), (2,0.25), (3,0.16) and joining them we get

- y = 3
^{x}

x-2

-1

0

1

2

3

^{x}3

^{-2}3

^{-1}3

^{0}3

^{1}3

^{2}y

0.1

0.3

1

3

9

Plot the points (-2,0.1), (-1,0.3), (0,1), (1,3), (2,9) and join to form a curve.

- y = log3
^{x}x1/9

1/3

1

3

9

log3

^{x}log

_{3}3^{-2}log

_{3}3^{-1}log

_{3}3^{0}log3

^{3}log

_{3}3^{2}y

-2

-1

0

1

2

Plot the points (1/9, -2), (1/3, -1), (1, 0), (3, 1), (9, 2) and join to form a curve.

- Absolute Values
- Adding and Subtracting Fractions
- Addition of Decimals
- Algebra Linear Equations
- Algebra Quadratic Equations
- Algebra Simultaneous Equations
- Algebraic Properties
- Algebraic Function
- Analyzing and Integrating
- Asymptotes
- Bar Graphs
- Basics of Statistics
- Circular Permutations
- Combinations
- Complex Numbers
- Complex Numbers AddSub
- Complex Numbers Division
- ComplexNumbers Multiplication
- Complex Numbers Properties
- Composite Functions
- Cube and Cube Roots
- Data collection Add Multipli Rules
- Datacollection GroupedMean
- Datacollection Mean
- Datacollection Median
- Datacollection Mode
- Datacollection Probabilitybasics
- Datacollection Probabilityevents
- Dividing Rational Numbers
- Division of Decimals
- Domain of SquareRootFunction
- EquationsReducibleQuadratic
- Exponential Functions
- ExponentialLogarithmicFunction
- Factorization
- FactorizationAnyQPolynomial
- FactorizationMonicQPolynomial
- Fractions & Decimal Conversion
- Functions
- Geometry Basics
- Graph of Rational Functions
- Graph of SquareRootFunction
- GraphicalRepresentation
- Graph of Complex Numbers
- Graphs Functions
- HighestCommonFactor
- Inequalities
- Inverse Square Functions
- Inverse of a Function
- Justifying Solutions
- LeastCommonMultiple
- Linear Equations2Variables
- LinearEquations3Variables
- Linear Equation System
- Linear Equation Graphs
- Linear Inequalities Graphs
- Maxima Minima Zeros
- More Functions
- Multiplication of Decimals
- Multiplying Rational Numbers
- Multiplying Two Polynomials
- Other Functions
- Permutations
- Pictorial BarChart
- Pictorial Bivariatedata
- Pictorial BoxPlots
- Pictorial FrequencyTable
- Pictorial Histogram
- Pictorial LineCharts
- Pictorial PieChart
- Pictorial StemLeafPlot
- Polynomials Addition
- Polynomials Division
- Polynomials Multiplication
- Predicting Values
- Problem Solving Strategies
- Quadratic Equation
- QuadraticEquationsFormula
- QuadraticEquationsSolutions
- QuadraticInequalities
- Rational Behind Functions
- Rational Expression
- Rational Functions
- Real Numbers
- Reciprocal Functions
- Recursive Multiplication
- Rational Numbers
- Review of functions
- Review of Sets and Relations
- Rounding Numbers
- Scientific Notation
- Simple Probabilities
- SimplifyingRationalExpressions
- SolutionQuadratEquawhen
- Solving Fractions
- SolvingQuadraticEquations
- Square of a Binomial
- SquareRootFunctions
- SquareRootFunInequalities
- Squares and Square Roots
- Step Function
- Subtraction of Decimals
- Types of Functions
- Unit Conversions Measurements
- WordProblemsQE