The table below is for the function of y = 2x² - x - 6. a. Complete the table. b. On the grid provided, plot the graph of y = 2x² - x - 6 for -3 ≤ x ≤ 4

Hapa kuna suluhisho la swali lako:

a) Kamilisha jedwali

Tunatumia kazi $y = 2x^2 - x - 6$ kukokotoa maadili ya $y$ kwa kila thamani ya $x$.

• Kwa $x = -3$: $y = 2(-3)^2 - (-3) - 6 = 2(9) + 3 - 6 = 18 + 3 - 6 = 15$
• Kwa $x = -2$: $y = 2(-2)^2 - (-2) - 6 = 2(4) + 2 - 6 = 8 + 2 - 6 = 4$
• Kwa $x = -1$: $y = 2(-1)^2 - (-1) - 6 = 2(1) + 1 - 6 = 2 + 1 - 6 = -3$
• Kwa $x = 0$: $y = 2(0)^2 - (0) - 6 = 0 - 0 - 6 = -6$
• Kwa $x = 1$: $y = 2(1)^2 - (1) - 6 = 2(1) - 1 - 6 = 2 - 1 - 6 = -5$
• Kwa $x = 2$: $y = 2(2)^2 - (2) - 6 = 2(4) - 2 - 6 = 8 - 2 - 6 = 0$
• Kwa $x = 3$: $y = 2(3)^2 - (3) - 6 = 2(9) - 3 - 6 = 18 - 3 - 6 = 9$
• Kwa $x = 4$: $y = 2(4)^2 - (4) - 6 = 2(16) - 4 - 6 = 32 - 4 - 6 = 22$

Jedwali lililokamilika ni: | $x$ | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---|---|---|---| | $y$ | 15 | 4 | -3 | -6 | -5 | 0 | 9 | 22 |

b) Chora grafu ya $y = 2x^2 - x - 6$ kwa $-3 \le x \le 4$

Tumia pointi zifuatazo kutoka kwenye jedwali ili kuchora grafu kwenye gridi uliyopewa: $(-3, 15)$, $(-2, 4)$, $(-1, -3)$, $(0, -6)$, $(1, -5)$, $(2, 0)$, $(3, 9)$, $(4, 22)$. Unganisha pointi hizi kwa kutumia mstari laini ili kuunda parabola.

c) Tumia grafu yako kutatua $2x^2 - x - 6 = 0$

Kutatua $2x^2 - x - 6 = 0$ kwa kutumia grafu kunamaanisha kutafuta maadili ya $x$ ambapo $y = 0$. Haya ni maeneo ambapo grafu inapita kwenye mhimili wa $x$. Kutoka kwenye jedwali na grafu, tunaona kwamba $y=0$ wakati $x=2$. Pia, grafu inapita mhimili wa $x$ kati ya $x=-2$ na $x=-1$. Kwa kukadiria kutoka kwenye grafu, au kwa kutatua mlinganyo, tunapata thamani nyingine. Kutatua mlinganyo $2x^2 - x - 6 = 0$: $(2x+3)(x-2) = 0$ Hivyo, $2x+3 = 0 \Rightarrow x = -\frac{3}{2} = -1.5$ Au $x-2 = 0 \Rightarrow x = 2$

Kwa hiyo, suluhisho kutoka kwenye grafu ni: $x = -1.5 \text{ na 2}$

3 done, 2 left today. You're making progress.