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 chaPTer 7 FUNCTIONS AND GRAPHS 237
7 
Functions and graphs
In this chapter, you will: 
•	 use the point-by-point plotting method for sketching the 
graph of an equation in two variables 
•	 with this method, you construct a table of values that 
consists of several solution points of the equation and then 
plot the solution points on the Cartesian plane
2153 TechMaths Eng G10 LB.indb   237 2015/10/22   3:41 PM
Page 2


 chaPTer 7 FUNCTIONS AND GRAPHS 237
7 
Functions and graphs
In this chapter, you will: 
•	 use the point-by-point plotting method for sketching the 
graph of an equation in two variables 
•	 with this method, you construct a table of values that 
consists of several solution points of the equation and then 
plot the solution points on the Cartesian plane
2153 TechMaths Eng G10 LB.indb   237 2015/10/22   3:41 PM
238 Technical Ma TheMaTicS Grade 10
7.1 Function notation
Function notation is one of many other conventions that we use to represent and name 
relationships between the values of two variables.
The definition of a function consists of the following components:
•	 the name of the function
•	 the variable that represents a value to evaluate the function
•	 a rule that says how to calculate the function's output for the given input value
The number that results from applying the rule to a specific input value is called an output 
value and is represented by f (x).
Consider the function given: f (x) = 3x + 2
 3x + 2 is a rule that says how to calculate
 the function’s output for a given input
  x is the variable that represents a value at which to 
  evaluate the function
 the letter f is the name of the function
7.2 revision of linear functions
Work through the examples to revise what you have learned about linear functions.
w orked examples 
A. Problem: 
 (a) Copy and complete the table below for the functions defined by f (x) = x.
f (-5) = -5 f (-4) = -4 f (-3) = -3 f (-2) = -2 f (-1) = -1 f (0) = 0
f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (5) = 5
 Solution:
(a) x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -5 -4 -3 -2 -1 0 1 2 3 4 5
2153 TechMaths Eng G10 LB.indb   238 2015/10/22   3:41 PM
Page 3


 chaPTer 7 FUNCTIONS AND GRAPHS 237
7 
Functions and graphs
In this chapter, you will: 
•	 use the point-by-point plotting method for sketching the 
graph of an equation in two variables 
•	 with this method, you construct a table of values that 
consists of several solution points of the equation and then 
plot the solution points on the Cartesian plane
2153 TechMaths Eng G10 LB.indb   237 2015/10/22   3:41 PM
238 Technical Ma TheMaTicS Grade 10
7.1 Function notation
Function notation is one of many other conventions that we use to represent and name 
relationships between the values of two variables.
The definition of a function consists of the following components:
•	 the name of the function
•	 the variable that represents a value to evaluate the function
•	 a rule that says how to calculate the function's output for the given input value
The number that results from applying the rule to a specific input value is called an output 
value and is represented by f (x).
Consider the function given: f (x) = 3x + 2
 3x + 2 is a rule that says how to calculate
 the function’s output for a given input
  x is the variable that represents a value at which to 
  evaluate the function
 the letter f is the name of the function
7.2 revision of linear functions
Work through the examples to revise what you have learned about linear functions.
w orked examples 
A. Problem: 
 (a) Copy and complete the table below for the functions defined by f (x) = x.
f (-5) = -5 f (-4) = -4 f (-3) = -3 f (-2) = -2 f (-1) = -1 f (0) = 0
f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (5) = 5
 Solution:
(a) x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -5 -4 -3 -2 -1 0 1 2 3 4 5
2153 TechMaths Eng G10 LB.indb   238 2015/10/22   3:41 PM
 chaPTer 7 FUNCTIONS AND GRAPHS 239
(b) Draw the graph of the function by 
plotting the coordinates on the 
Cartesian plane.
(c) The x-intercept: (0; 0)
(d) The y-intercept: (0 0)
(e) Domain: x ? R
(f) Range: y ? R
x
y
–14 –12 –10 –6 –4 –2 2 0 4 6 8 10 12 14 –8
10
8
6
4
–2
–4
–6
–8
–10
2
w orked example 
B. Problem: A graph of a certain function is given below. 
x
y
–6 –4 –2 2 0 4 6 8 –8
12
10
8
6
4
–2
–4
–6
–8
–10
2
Copy and complete the table for this function by reading the values from the graph.
Solution:
x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -8 -6 -4 -2 0 2 4 6 8 10 12
(x; f (x)) (-5; -8) (-4; -6) (-3; -4) (-2; -2) (-1; 0) (0; 2) (1; 4) (2; 6) (3; 8) (4; 10) (5; 12)
(b) x-intercept: (-1; 0) (c) y-intercept: (0; 2)
(d) Domain: x ? R  (e) Range: y ? R 
The domain is the set of all possible input values to which the rule applies.
The range is the set of all possible output values to which the rule applies.
2153 TechMaths Eng G10 LB.indb   239 2015/10/22   3:41 PM
Page 4


 chaPTer 7 FUNCTIONS AND GRAPHS 237
7 
Functions and graphs
In this chapter, you will: 
•	 use the point-by-point plotting method for sketching the 
graph of an equation in two variables 
•	 with this method, you construct a table of values that 
consists of several solution points of the equation and then 
plot the solution points on the Cartesian plane
2153 TechMaths Eng G10 LB.indb   237 2015/10/22   3:41 PM
238 Technical Ma TheMaTicS Grade 10
7.1 Function notation
Function notation is one of many other conventions that we use to represent and name 
relationships between the values of two variables.
The definition of a function consists of the following components:
•	 the name of the function
•	 the variable that represents a value to evaluate the function
•	 a rule that says how to calculate the function's output for the given input value
The number that results from applying the rule to a specific input value is called an output 
value and is represented by f (x).
Consider the function given: f (x) = 3x + 2
 3x + 2 is a rule that says how to calculate
 the function’s output for a given input
  x is the variable that represents a value at which to 
  evaluate the function
 the letter f is the name of the function
7.2 revision of linear functions
Work through the examples to revise what you have learned about linear functions.
w orked examples 
A. Problem: 
 (a) Copy and complete the table below for the functions defined by f (x) = x.
f (-5) = -5 f (-4) = -4 f (-3) = -3 f (-2) = -2 f (-1) = -1 f (0) = 0
f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (5) = 5
 Solution:
(a) x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -5 -4 -3 -2 -1 0 1 2 3 4 5
2153 TechMaths Eng G10 LB.indb   238 2015/10/22   3:41 PM
 chaPTer 7 FUNCTIONS AND GRAPHS 239
(b) Draw the graph of the function by 
plotting the coordinates on the 
Cartesian plane.
(c) The x-intercept: (0; 0)
(d) The y-intercept: (0 0)
(e) Domain: x ? R
(f) Range: y ? R
x
y
–14 –12 –10 –6 –4 –2 2 0 4 6 8 10 12 14 –8
10
8
6
4
–2
–4
–6
–8
–10
2
w orked example 
B. Problem: A graph of a certain function is given below. 
x
y
–6 –4 –2 2 0 4 6 8 –8
12
10
8
6
4
–2
–4
–6
–8
–10
2
Copy and complete the table for this function by reading the values from the graph.
Solution:
x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -8 -6 -4 -2 0 2 4 6 8 10 12
(x; f (x)) (-5; -8) (-4; -6) (-3; -4) (-2; -2) (-1; 0) (0; 2) (1; 4) (2; 6) (3; 8) (4; 10) (5; 12)
(b) x-intercept: (-1; 0) (c) y-intercept: (0; 2)
(d) Domain: x ? R  (e) Range: y ? R 
The domain is the set of all possible input values to which the rule applies.
The range is the set of all possible output values to which the rule applies.
2153 TechMaths Eng G10 LB.indb   239 2015/10/22   3:41 PM
240 Technical Ma TheMaTicS Grade 10
exercises
1 Copy and complete the table for the functions defined by:
 (a) f (x) = -x (b) g (x) = x + 5 (c) h (x) = -x - 1
x -5 -4 -3 -2 -1 0 1 2 3 4 5 6
f (x)
g (x)
h (x)
2 Draw the graphs of each function in exercise 1 above on the same set of axes. 
3 Complete the table below for the three functions drawn in exercise 2.
Function x-intercept y-intercept Domain r ange
f (x) = -x
g (x) = x + 5
h (x) = -x - 1
4 Use the intercept method to sketch the graphs of the following functions:
 (a) y = x + 2 (b) y = x - 3 (c) y = 2x + 1  (d) y = -2x + 4
 (e) x = y - 4 (f) x = y (g) y = -3x + 6
5  A graph of a certain function g (x) is given below. 
 Copy the table below for the function 
g (x). By reading of corresponding x- and 
y-values of points on the graph of g (x), 
complete the table by writing down 
specific corresponding x- and y-values.
x
y
–5 –6 –3 –2 –1 1 0 2 3 4 5 6 7 –4
6
7
5
4
3
2
–1
–2
–3
–4
–5
–6
–7
1
x
y = g (x)
2153 TechMaths Eng G10 LB.indb   240 2015/10/22   3:41 PM
Page 5


 chaPTer 7 FUNCTIONS AND GRAPHS 237
7 
Functions and graphs
In this chapter, you will: 
•	 use the point-by-point plotting method for sketching the 
graph of an equation in two variables 
•	 with this method, you construct a table of values that 
consists of several solution points of the equation and then 
plot the solution points on the Cartesian plane
2153 TechMaths Eng G10 LB.indb   237 2015/10/22   3:41 PM
238 Technical Ma TheMaTicS Grade 10
7.1 Function notation
Function notation is one of many other conventions that we use to represent and name 
relationships between the values of two variables.
The definition of a function consists of the following components:
•	 the name of the function
•	 the variable that represents a value to evaluate the function
•	 a rule that says how to calculate the function's output for the given input value
The number that results from applying the rule to a specific input value is called an output 
value and is represented by f (x).
Consider the function given: f (x) = 3x + 2
 3x + 2 is a rule that says how to calculate
 the function’s output for a given input
  x is the variable that represents a value at which to 
  evaluate the function
 the letter f is the name of the function
7.2 revision of linear functions
Work through the examples to revise what you have learned about linear functions.
w orked examples 
A. Problem: 
 (a) Copy and complete the table below for the functions defined by f (x) = x.
f (-5) = -5 f (-4) = -4 f (-3) = -3 f (-2) = -2 f (-1) = -1 f (0) = 0
f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (5) = 5
 Solution:
(a) x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -5 -4 -3 -2 -1 0 1 2 3 4 5
2153 TechMaths Eng G10 LB.indb   238 2015/10/22   3:41 PM
 chaPTer 7 FUNCTIONS AND GRAPHS 239
(b) Draw the graph of the function by 
plotting the coordinates on the 
Cartesian plane.
(c) The x-intercept: (0; 0)
(d) The y-intercept: (0 0)
(e) Domain: x ? R
(f) Range: y ? R
x
y
–14 –12 –10 –6 –4 –2 2 0 4 6 8 10 12 14 –8
10
8
6
4
–2
–4
–6
–8
–10
2
w orked example 
B. Problem: A graph of a certain function is given below. 
x
y
–6 –4 –2 2 0 4 6 8 –8
12
10
8
6
4
–2
–4
–6
–8
–10
2
Copy and complete the table for this function by reading the values from the graph.
Solution:
x -5 -4 -3 -2 -1 0 1 2 3 4 5
f (x) -8 -6 -4 -2 0 2 4 6 8 10 12
(x; f (x)) (-5; -8) (-4; -6) (-3; -4) (-2; -2) (-1; 0) (0; 2) (1; 4) (2; 6) (3; 8) (4; 10) (5; 12)
(b) x-intercept: (-1; 0) (c) y-intercept: (0; 2)
(d) Domain: x ? R  (e) Range: y ? R 
The domain is the set of all possible input values to which the rule applies.
The range is the set of all possible output values to which the rule applies.
2153 TechMaths Eng G10 LB.indb   239 2015/10/22   3:41 PM
240 Technical Ma TheMaTicS Grade 10
exercises
1 Copy and complete the table for the functions defined by:
 (a) f (x) = -x (b) g (x) = x + 5 (c) h (x) = -x - 1
x -5 -4 -3 -2 -1 0 1 2 3 4 5 6
f (x)
g (x)
h (x)
2 Draw the graphs of each function in exercise 1 above on the same set of axes. 
3 Complete the table below for the three functions drawn in exercise 2.
Function x-intercept y-intercept Domain r ange
f (x) = -x
g (x) = x + 5
h (x) = -x - 1
4 Use the intercept method to sketch the graphs of the following functions:
 (a) y = x + 2 (b) y = x - 3 (c) y = 2x + 1  (d) y = -2x + 4
 (e) x = y - 4 (f) x = y (g) y = -3x + 6
5  A graph of a certain function g (x) is given below. 
 Copy the table below for the function 
g (x). By reading of corresponding x- and 
y-values of points on the graph of g (x), 
complete the table by writing down 
specific corresponding x- and y-values.
x
y
–5 –6 –3 –2 –1 1 0 2 3 4 5 6 7 –4
6
7
5
4
3
2
–1
–2
–3
–4
–5
–6
–7
1
x
y = g (x)
2153 TechMaths Eng G10 LB.indb   240 2015/10/22   3:41 PM
 chaPTer 7 FUNCTIONS AND GRAPHS 241
7.3 Quadratic function y =  ax 
2
 
w orked example
Problem: Consider the function defined by f (x) = x
2
.
Solution:
Step 1: Calculate the output values of f (x) = x
2
 for integer x-values -4 to +4.
f (-3) = (-3 ) 
2
  = 9 f (0) = (0 ) 
2
  = 0 f (3) = (3 ) 
2
  = 9
f (-2) = (-2 ) 
2
  = 4 f (1) = (1 ) 
2
  = 1 f (4) = (4 ) 
2
  = 1 6
Step 2: Represent the function f(x) = x
2
 
by means of a table.
x -4 -3 -2 -1 0 1 2 3 4
f (x) 16 9 4 1 0 1 4 9 16
Step 3: Draw the graph representing the function f (x) = x
2
.
x
y
–6 –8 –4 –2 2 0 4 6 8
12
14
16
10
8
6
4
–2
–4
2
Step 4: Identify the shape of the graph.
  The graph opens upwards (shape).
Step 5: Identify the intercepts of the graph.
  The graph intercepts the x-axis at one point (0; 0).
  The x-intercepts are also called the roots.
An intercept is a point at which the graph 
intersects the x- or y-axis.
2153 TechMaths Eng G10 LB.indb   241 2015/10/22   3:41 PM

	Mathematics for Grade 10 34 docs

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FAQs on Textbook: Functions and Graphs - Mathematics for Grade 10

1. What are the different types of functions that we study in Grade 10?

Ans. In Grade 10, we typically study several types of functions, including linear functions, quadratic functions, exponential functions, and rational functions. Each type has its unique characteristics and graphs. Linear functions have a constant rate of change, quadratic functions form a parabola, exponential functions grow rapidly, and rational functions are ratios of polynomials.

2. How can I graph a linear function?

Ans. To graph a linear function, first, identify the slope and y-intercept from the function's equation in slope-intercept form (y = mx + b). The slope (m) indicates the steepness of the line, and the y-intercept (b) is the point where the line crosses the y-axis. Start by plotting the y-intercept on the graph, then use the slope to find another point by rising and running from the y-intercept. Connect these points to form a straight line.

3. What is the importance of the vertex in quadratic functions?

Ans. The vertex of a quadratic function is the highest or lowest point on its graph, depending on whether it opens upwards or downwards. It represents the maximum or minimum value of the function. Understanding the vertex helps in analyzing the function's behavior, finding its maximum or minimum value, and solving optimization problems.

4. How do you determine if a function is increasing or decreasing?

Ans. To determine if a function is increasing or decreasing, you can analyze its graph or its derivative (if applicable). A function is increasing on an interval if the y-values rise as you move from left to right, while it is decreasing if the y-values fall. For linear functions, this is straightforward; for more complex functions, examining the slope or the first derivative can provide insight into where the function is increasing or decreasing.

5. What are the key features to identify in the graph of a rational function?

Ans. When analyzing the graph of a rational function, key features to identify include vertical and horizontal asymptotes, intercepts (x and y), and any holes in the graph. Vertical asymptotes occur where the denominator is zero, while horizontal asymptotes indicate the behavior of the function as x approaches infinity. Intercepts can be found by setting the numerator and denominator to zero, and holes occur at points where both the numerator and denominator are zero.

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