1. **Stating the problem:** We have a table of values for $x$ and $y$: $$\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ y & 5 & 7 & 9 & _ & _ & _ & 17 \\\end{array}$$ We need to: - (ii) Determine the rule for the mapping from $x$ to $y$. - (b) Plot the points for the ordered pairs $(x,y)$. - (c) Find: - (i) $y$ when $x=0$. - (ii) $x$ when $y=14$. 2. **Determine the rule for the mapping:** Look at the given $y$ values for $x=1,2,3$: - When $x=1$, $y=5$ - When $x=2$, $y=7$ - When $x=3$, $y=9$ Notice the pattern: $y$ increases by 2 as $x$ increases by 1. This suggests a linear rule of the form: $$y = mx + c$$ where $m$ is the slope and $c$ is the $y$-intercept. Calculate the slope $m$: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{2 - 1} = \frac{2}{1} = 2$$ Use one point to find $c$: $$5 = 2(1) + c \implies c = 5 - 2 = 3$$ So the rule is: $$y = 2x + 3$$ 3. **Find missing $y$ values for $x=4,5,6$:** - For $x=4$: $$y = 2(4) + 3 = 8 + 3 = 11$$ - For $x=5$: $$y = 2(5) + 3 = 10 + 3 = 13$$ - For $x=6$: $$y = 2(6) + 3 = 12 + 3 = 15$$ Check for $x=7$: $$y = 2(7) + 3 = 14 + 3 = 17$$ which matches the given value. 4. **Plot the points:** The ordered pairs are: $$(1,5), (2,7), (3,9), (4,11), (5,13), (6,15), (7,17)$$ 5. **Find $y$ when $x=0$:** Using the rule: $$y = 2(0) + 3 = 3$$ 6. **Find $x$ when $y=14$:** Set $y=14$ and solve for $x$: $$14 = 2x + 3$$ $$2x = 14 - 3 = 11$$ $$x = \frac{11}{2} = 5.5$$ **Final answers:** - Rule: $y = 2x + 3$ - Missing $y$ values: 11, 13, 15 - $y$ when $x=0$ is 3 - $x$ when $y=14$ is 5.5