A student is investigating linear graphs. They have completed a table showing the gradients and y-intercepts of three linear equations and have plotted the graphs. The student needs help writing up th

2.1:
The standard equation of a linear graph is given by the slope-intercept form.

$$y = mx + c$$

where $m$ is the gradient (slope) and $c$ is the y-intercept.

2.2.1:
To complete the table, we need three linear equations. The problem provides one row with a gradient of 2 and a y-intercept of 0, which corresponds to the equation $y = 2x$. We will assume this is the "mother graph" for investigation. For the other two rows, we will choose variations to investigate the effect of changing the gradient and the y-intercept.

Let's choose:

• A graph with a different gradient but the same y-intercept as $y=2x$. For example, $y=x$.
• A graph with the same gradient but a different y-intercept as $y=2x$. For example, $y=2x+1$.

Completing the table:

Equation	Gradient	Y-intercept
$y=x$	1	0
$y=2x$	2	0
$y=2x+1$	2	1

2.2.2:
Based on the completed table, the gradients of the 3 linear graphs are:
1, 2, 2

2.2.3:
To plot the three graphs, we find a few points for each equation:

1. For $y=x$:

   • If $x=0$, $y=0$
   • If $x=2$, $y=2$
   • If $x=-2$, $y=-2$ Plot points $(0,0)$, $(2,2)$, $(-2,-2)$ and draw a straight line through them.

2. For $y=2x$:

   • If $x=0$, $y=0$
   • If $x=1$, $y=2$
   • If $x=-1$, $y=-2$ Plot points $(0,0)$, $(1,2)$, $(-1,-2)$ and draw a straight line through them.

3. For $y=2x+1$:

   • If $x=0$, $y=1$
   • If $x=1$, $y=3$
   • If $x=-1$, $y=-1$ Plot points $(0,1)$, $(1,3)$, $(-1,-1)$ and draw a straight line through them.

The completed table and the gradients are:

• 2.1: $\boxed{y = mx + c}$
• 2.2.1:

  Equation	Gradient	Y-intercept
  $y=x$	1	0
  $y=2x$	2	0
  $y=2x+1$	2	1

• 2.2.2: The gradients of the 3 linear graphs are $\boxed{1, 2, 2}$.
• 2.2.3: Plot the lines $y=x$, $y=2x$, and $y=2x+1$ on the provided Cartesian plane using the points calculated above.