Given the linear equations below: a) y = 3x + 1 b) y = -2x + 2 a. From the equations, derive the value of the gradient and y-intercept. b. Plot the graph of on a common axis. c. Is the function increasing, decreasing or constant?

The equation is in the form $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.
Comparing $y = 3x + 1$ with $y = mx + c$:

• The gradient $m$ is the coefficient of $x$.
• The y-intercept $c$ is the constant term.

$$m = 3$$ $$c = 1$$

The y-intercept is the point $(0, c)$, which is $(0, 1)$.

To plot the graph, we need at least two points. We already have the y-intercept $(0, 1)$. Let's find another point by choosing an $x$-value, for example, $x=1$.
When $x = 1$:

$$y = 3(1) + 1$$ $$y = 3 + 1$$ $$y = 4$$

So, another point is $(1, 4)$.
To plot the graph, draw a Cartesian plane, mark the points $(0, 1)$ and $(1, 4)$, and draw a straight line passing through them.

The nature of the function is determined by the sign of the gradient.

• If $m > 0$, the function is increasing.
• If $m < 0$, the function is decreasing.
• If $m = 0$, the function is constant. Since the gradient $m = 3$, which is greater than 0, the function is increasing.

For equation b) $y = -2x + 2$

Comparing $y = -2x + 2$ with $y = mx + c$:

• The gradient $m$ is the coefficient of $x$.
• The y-intercept $c$ is the constant term.

$$m = -2$$ $$c = 2$$

The y-intercept is the point $(0, c)$, which is $(0, 2)$.

We have the y-intercept $(0, 2)$. Let's find another point by choosing an $x$-value, for example, $x=1$.
When $x = 1$:

$$y = -2(1) + 2$$ $$y = -2 + 2$$ $$y = 0$$

So, another point is $(1, 0)$.
To plot the graph, draw a Cartesian plane, mark the points $(0, 2)$ and $(1, 0)$, and draw a straight line passing through them.

Since the gradient $m = -2$, which is less than 0, the function is decreasing.

Here's a summary of the answers:

1. From the equations above, state the value of:

a) the gradient and y-intercept
For $y = 3x + 1$:
Gradient: $\boxed{3}$
Y-intercept: $\boxed{1}$

For $y = -2x + 2$:
Gradient: $\boxed{-2}$
Y-intercept: $\boxed{2}$

b) Plot the graph of each equation on a Cartesian plane.
For $y = 3x + 1$:
Points for plotting: $\boxed{(0, 1) \text{ and } (1, 4)}$

For $y = -2x + 2$:
Points for plotting: $\boxed{(0, 2) \text{ and } (1, 0)}$

c) Is the function increasing, decreasing or constant?
For $y = 3x + 1$:
The function is $\boxed{increasing}$.

For $y = -2x + 2$:
The function is $\boxed{decreasing}$.