Question Statement

Plot the graph of the following functions and find the points of intersection of each function with the axes:

(i) $y = x + 3$ (ii) $y = 6 - x$ (iii) $y = x^{2} - 5 x$

Background and Explanation

To plot a graph, we select various values for $x$ and calculate the corresponding $y$ values to create coordinates. The points of intersection with the axes are the $y$ -intercept (where $x = 0$ ) and the $x$ -intercept (where $y = 0$ ).

Solution

Part (i) $y = x + 3$

To plot this linear function, we can calculate a few points:

$x$	1	-1
$y$	4	2

By plotting these points and drawing a straight line through them, we obtain the following graph:

Finding Intersections:

$y$ -intercept: Set $x = 0$ , then $y = 0 + 3 = 3$ . The point is $(0, 3)$ .
$x$ -intercept: Set $y = 0$ , then $0 = x + 3 ⟹ x = - 3$ . The point is $(- 3, 0)$ .

From the graph, it is clear that the $x$ -intercept is $- 3$ and the $y$ -intercept is $3$ .

Part (ii) $y = 6 - x$

We determine the coordinates by choosing values for $x$ :

$x$	0	6
$y$	6	0

Plotting these points results in the following graph:

Finding Intersections:

When $x = 0$ , $y = 6 - 0 = 6$ .
When $y = 0$ , $0 = 6 - x ⟹ x = 6$ .

The points of intersection of the function with the axes are $(0, 6)$ and $(6, 0)$ .

Part (iii) $y = x^{2} - 5 x$

This is a quadratic function (a parabola). We calculate the intercepts and the vertex to plot it:

$x$	0	5	2.5
$y$	0	0	-6.25

Plotting these points gives us the following curve:

Finding Intersections:

$y$ -intercept: Set $x = 0$ , then $y = 0^{2} - 5 (0) = 0$ . The point is $(0, 0)$ .
$x$ -intercepts: Set $y = 0$ , then $0 = x^{2} - 5 x ⟹ 0 = x (x - 5)$ . This gives $x = 0$ and $x = 5$ .

The points of intersection of the function with the axes are $(0, 0)$ and $(5, 0)$ .

Key Formulas or Methods Used

Table of Values: Choosing independent values ( $x$ ) to find dependent values ( $y$ ).
$y$ -intercept: Calculated by setting $x = 0$ .
$x$ -intercept: Calculated by setting $y = 0$ .
Linear Equations: Forms a straight line ( $y = m x + c$ ).
Quadratic Equations: Forms a parabola ( $y = a x^{2} + b x + c$ ).

Summary of Steps

Create a table of values by substituting specific $x$ values into the given equation.
Plot the resulting $(x, y)$ coordinates on a Cartesian plane.
Connect the points (using a straight edge for linear functions and a smooth curve for quadratics).
Identify the points where the graph crosses the horizontal $x$ -axis and the vertical $y$ -axis.

Question Statement

Plot the graph of the following functions and find the points of intersection of each function with the axes:

(i) $y = x + 3$ (ii) $y = 6 - x$ (iii) $y = x^{2} - 5 x$

Background and Explanation

Solution

Part (i) $y = x + 3$

To plot this linear function, we can calculate a few points:

$x$	1	-1
$y$	4	2

By plotting these points and drawing a straight line through them, we obtain the following graph:

Finding Intersections:

$y$ -intercept: Set $x = 0$ , then $y = 0 + 3 = 3$ . The point is $(0, 3)$ .
$x$ -intercept: Set $y = 0$ , then $0 = x + 3 ⟹ x = - 3$ . The point is $(- 3, 0)$ .

From the graph, it is clear that the $x$ -intercept is $- 3$ and the $y$ -intercept is $3$ .

Part (ii) $y = 6 - x$

We determine the coordinates by choosing values for $x$ :

$x$	0	6
$y$	6	0

Plotting these points results in the following graph:

Finding Intersections:

When $x = 0$ , $y = 6 - 0 = 6$ .
When $y = 0$ , $0 = 6 - x ⟹ x = 6$ .

The points of intersection of the function with the axes are $(0, 6)$ and $(6, 0)$ .

Part (iii) $y = x^{2} - 5 x$

This is a quadratic function (a parabola). We calculate the intercepts and the vertex to plot it:

$x$	0	5	2.5
$y$	0	0	-6.25

Plotting these points gives us the following curve:

Finding Intersections:

$y$ -intercept: Set $x = 0$ , then $y = 0^{2} - 5 (0) = 0$ . The point is $(0, 0)$ .
$x$ -intercepts: Set $y = 0$ , then $0 = x^{2} - 5 x ⟹ 0 = x (x - 5)$ . This gives $x = 0$ and $x = 5$ .

The points of intersection of the function with the axes are $(0, 0)$ and $(5, 0)$ .

Key Formulas or Methods Used

Table of Values: Choosing independent values ( $x$ ) to find dependent values ( $y$ ).
$y$ -intercept: Calculated by setting $x = 0$ .
$x$ -intercept: Calculated by setting $y = 0$ .
Linear Equations: Forms a straight line ( $y = m x + c$ ).
Quadratic Equations: Forms a parabola ( $y = a x^{2} + b x + c$ ).

Summary of Steps

Create a table of values by substituting specific $x$ values into the given equation.
Plot the resulting $(x, y)$ coordinates on a Cartesian plane.
Connect the points (using a straight edge for linear functions and a smooth curve for quadratics).
Identify the points where the graph crosses the horizontal $x$ -axis and the vertical $y$ -axis.

1.2 Q-6

Question Statement

Background and Explanation

Solution

Part (i) $y = x + 3$

Part (ii) $y = 6 - x$

Part (iii) $y = x^{2} - 5 x$

Key Formulas or Methods Used

Summary of Steps

1.2 Q-6

Question Statement

Background and Explanation

Solution

Part (i) $y = x + 3$

Part (ii) $y = 6 - x$

Part (iii) $y = x^{2} - 5 x$

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Part (i) y=x+3

Part (ii) y=6−x

Part (iii) y=x2−5x

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Part (i) y=x+3

Part (ii) y=6−x

Part (iii) y=x2−5x

Key Formulas or Methods Used

Summary of Steps

Part (i) $y = x + 3$

Part (ii) $y = 6 - x$

Part (iii) $y = x^{2} - 5 x$

Part (i) $y = x + 3$

Part (ii) $y = 6 - x$

Part (iii) $y = x^{2} - 5 x$