Given the mapping x: 1, 2, 3, 4, 5 and y: 3, 8, 15, 24, 35, find the rule for the mapping.

The given mapping is:
$x$: 1, 2, 3, 4, 5
$y$: 3, 8, 15, 24, 35

Calculate the first differences in $y$ values:
$8 - 3 = 5$
$15 - 8 = 7$
$24 - 15 = 9$
$35 - 24 = 11$

The first differences are 5, 7, 9, 11.

$7 - 5 = 2$
$9 - 7 = 2$
$11 - 9 = 2$

Since the second differences are constant and equal to 2, the rule is a quadratic function of the form $y = ax^2 + bx + c$.
The second difference is equal to $2a$.

$$2a = 2$$ $$a = 1$$

The equation becomes $y = x^2 + bx + c$.

Using the point $(x=1, y=3)$:

$$3 = (1)^2 + b(1) + c$$ $$3 = 1 + b + c$$ $$b + c = 2 \quad (*)$$

Using the point $(x=2, y=8)$:

$$8 = (2)^2 + b(2) + c$$ $$8 = 4 + 2b + c$$ $$2b + c = 4 \quad (**)$$

Subtract equation $(*)$ from equation $(**)$:

$$(2b + c) - (b + c) = 4 - 2$$ $$b = 2$$

Substitute $b=2$ into equation $(*)$:

$$2 + c = 2$$ $$c = 0$$

Substitute $a=1$, $b=2$, and $c=0$ into $y = ax^2 + bx + c$:

$$y = 1x^2 + 2x + 0$$ $$y = x^2 + 2x$$