Explain the relationship between gradients and y-intercepts of linear graphs, calculate the product of gradients, and plot the graphs.

• 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.

The table provides two equations and their gradients:

• For $y = 2x + 1$, the gradient is $m_1 = 2$.
• For $y = -2x + 1$, the gradient is $m_2 = -2$.

The product of the two gradients is:

The product of the two gradients is equal to $\boxed{-4}$.

We need to plot $y = 2x + 1$ and $y = -2x + 1$.

• For the graph $y = 2x + 1$:

  • The y-intercept is $c = 1$, so the line passes through $(0, 1)$.
  • The gradient is $m = 2$. From the y-intercept $(0, 1)$, move 1 unit to the right and 2 units up to find another point, $(1, 3)$.
  • Draw a straight line through $(0, 1)$ and $(1, 3)$.

• For the graph $y = -2x + 1$:

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

Both lines intersect at the point $(0, 1)$.