1. **Stating the problem:** We have a scatter plot with points (1,7), (2,4), and (3,2) representing input-output pairs (Entrée, Sortie). We want to find the table of values and an expression that models this data. 2. **Table of values:** From the points, we can write the table: | Entrée (x) | Sortie (y) | |------------|------------| | 1 | 7 | | 2 | 4 | | 3 | 2 | 3. **Finding the expression:** We look for a function $y=f(x)$ that fits these points. Since the points decrease as $x$ increases, a linear function is a good start: $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 4. **Calculate the slope $m$:** Using points $(1,7)$ and $(2,4)$, $$m = \frac{4 - 7}{2 - 1} = \frac{-3}{1} = -3$$ 5. **Find $b$ using point $(1,7)$:** $$7 = -3(1) + b \implies b = 7 + 3 = 10$$ 6. **Write the expression:** $$y = -3x + 10$$ 7. **Verify with point $(3,2)$:** $$y = -3(3) + 10 = -9 + 10 = 1$$ This is close but not exactly 2, so the data is not perfectly linear. We can consider a quadratic expression: $$y = ax^2 + bx + c$$ 8. **Set up system with points:** For $(1,7)$: $a(1)^2 + b(1) + c = 7 \Rightarrow a + b + c = 7$ For $(2,4)$: $4a + 2b + c = 4$ For $(3,2)$: $9a + 3b + c = 2$ 9. **Solve the system:** Subtract first from second: $4a + 2b + c - (a + b + c) = 4 - 7 \Rightarrow 3a + b = -3$ Subtract first from third: $9a + 3b + c - (a + b + c) = 2 - 7 \Rightarrow 8a + 2b = -5$ 10. **From $3a + b = -3$, express $b = -3 - 3a$** Substitute into $8a + 2b = -5$: $$8a + 2(-3 - 3a) = -5 \Rightarrow 8a - 6 - 6a = -5 \Rightarrow 2a - 6 = -5 \Rightarrow 2a = 1 \Rightarrow a = \frac{1}{2}$$ 11. **Find $b$:** $$b = -3 - 3 \times \frac{1}{2} = -3 - \frac{3}{2} = -\frac{9}{2}$$ 12. **Find $c$ using $a + b + c = 7$:** $$\frac{1}{2} - \frac{9}{2} + c = 7 \Rightarrow -4 + c = 7 \Rightarrow c = 11$$ 13. **Final quadratic expression:** $$y = \frac{1}{2}x^2 - \frac{9}{2}x + 11$$ 14. **Verification:** At $x=3$: $$y = \frac{1}{2} \times 9 - \frac{9}{2} \times 3 + 11 = 4.5 - 13.5 + 11 = 2$$ Matches perfectly. **Answer:** Table: | Entrée (x) | Sortie (y) | |------------|------------| | 1 | 7 | | 2 | 4 | | 3 | 2 | Expression: $$y = \frac{1}{2}x^2 - \frac{9}{2}x + 11$$