Explain the relationship between the gradient and y-intercept of linear equations and their graphs. Also, calculate the product of gradients and plot the graphs for given equations.

When the gradient is the same, the graphs are parallel to each other.

As the y-intercept increases, the graph shifts upwards. Alternatively, as the y-intercept decreases, the graph shifts downwards.

When the gradient is positive, the graphs increases.

Complete the table by finding the equations:

The product of the two gradients is equal to:

To plot the graphs for $y = 2x - 2$ and $y = -2x + 1$:

• For $y = 2x - 2$:

  • The y-intercept is $c = -2$. Plot the point $(0, -2)$.
  • The gradient is $m = 2$. From $(0, -2)$, move 1 unit right and 2 units up to get to $(1, 0)$.
  • Draw a straight line through $(0, -2)$ and $(1, 0)$.

• For $y = -2x + 1$:

  • The y-intercept is $c = 1$. Plot the point $(0, 1)$.
  • The gradient is $m = -2$. From $(0, 1)$, move 1 unit right and 2 units down to get to $(1, -1)$.
  • Draw a straight line through $(0, 1)$ and $(1, -1)$.