Question

select the correct answer.
which equation best models the set of data shown in this table?

x	0	2	4	6	8	10	12	14	16	18
y	11	19	27	35	37	40	44	45	48	52

a. $y = 11\sqrt{x + 0.3} - 4.3$
b. $y = 11\sqrt{x - 0.3} + 4.3$
c. $y = 2x - 17$
d. $y = 2x + 17$

Explanation:

Step1: Test Option C and D with x=0

For Option C: \( y = 2(0)-17=-17 \), but table has y=11. Incorrect.
For Option D: \( y = 2(0)+17 = 17 \), close to 11? Wait, wait, x=2:

Step2: Test Option D with x=2

\( y = 2(2)+17=4 + 17=21 \)? No, table has y=19. Wait, maybe linear? Wait, first few points: x=0,y=11; x=2,y=19 (diff 8); x=4,y=27 (diff 8); x=6,y=35 (diff 8). Oh, first three differences are 8, so slope \( m=\frac{19 - 11}{2 - 0}=\frac{8}{2}=4 \)? Wait no, 19-11=8, 2-0=2, so 4? Wait no, wait 11 to 19 is +8 (x+2), 19 to 27 is +8 (x+2), 27 to 35 is +8 (x+2). So slope is \( \frac{8}{2}=4 \)? But options C and D have slope 2. Wait, maybe I miscalculated. Wait x=0,y=11; x=2,y=19: \( \frac{19 - 11}{2 - 0}=\frac{8}{2}=4 \). But options C and D have slope 2. Wait, maybe the later points are different. Wait x=8,y=37. From x=6,y=35: 37-35=2 (x+2). So the slope changes. So it's not linear. Wait, but let's check the square root options. Let's test Option B with x=0: \( y = 11\sqrt{0 - 0.3}+4.3 \), but square root of negative is invalid. Option A: \( y = 11\sqrt{0 + 0.3}-4.3 \approx 11*0.5477 - 4.3 \approx 6.02 - 4.3 = 1.72 \), no. Wait, maybe I made a mistake. Wait first three points: x=0,y=11; x=2,y=19; x=4,y=27; x=6,y=35. The pattern here is y = 4x + 11? Wait 4*0+11=11, 4*2+11=19, 4*4+11=27, 4*6+11=35. Yes! So that's linear with slope 4? But options don't have that. Wait, maybe the problem has a typo, or I misread. Wait the options: C is 2x-17, D is 2x+17. Wait x=0: D gives 17, A: 11*sqrt(0.3)-4.3≈1.7, B: invalid. Wait, maybe the first three points are linear with slope 4, but later points deviate. But the options given: let's check x=0 in Option D: 17, close to 11? No. Wait, maybe the question's table has a typo, or I miscalculate. Wait, wait the user's table: x=0,y=11; x=2,y=19; x=4,y=27; x=6,y=35; x=8,y=37. So from x=6 to x=8, y increases by 2 (35 to 37), x=8 to 10: 37 to 40 (+3), x=10 to 12: 40 to 44 (+4), etc. So the first part is linear with slope 4, then slope changes. But the options: let's check the square root options. Option B: x=0 is invalid (negative inside sqrt). Option A: x=0: \( 11\sqrt{0.3}-4.3≈11*0.5477 - 4.3≈6.02 - 4.3 = 1.72 \), no. Wait, maybe the intended answer is D, but with x=2: 2*2+17=21, but table has 19. Wait, maybe I made a mistake in slope. Wait 11 to 19: 8 over 2, so 4. But options have slope 2. Wait, maybe the problem is designed to have the first few points as linear with slope 4, but the options are wrong? No, maybe I misread the table. Wait the table: x=0,y=11; x=2,y=19; x=4,y=27; x=6,y=35. So 11,19,27,35: each time +8 when x+2. So the equation is y = 4x + 11. But that's not an option. Wait the options: C is 2x-17 (y=-17 when x=0, no), D is 2x+17 (y=17 when x=0, close to 11). Wait, maybe the question has a typo, and the first y is 17? Or maybe I miscalculate. Wait, let's check Option D with x=0: 17, x=2: 21, but table has 19. Option A: x=0: 11*sqrt(0.3)-4.3≈1.7, no. Option B: x=0: invalid. Wait, maybe the correct answer is D, but the table has a typo, or I'm wrong. Wait, maybe the slope is 2. Let's see: 2x + 11: x=0,y=11; x=2,y=15 (no, table has 19). 4x + 11: x=0,11; x=2,19 (yes!). But that's not an option. Wait the options given: maybe the problem is wrong, but among the options, let's check the square root. Option A: x=0: 11*sqrt(0.3)-4.3≈1.7, no. Option B: x=0: invalid. So maybe the intended answer is D, even though it's not perfect, or maybe I made a mistake. Wait, wait the first three points: 11,19,27,35: the difference is 8, so slope 4, but options have slope 2. So maybe the problem is designed to have the equation as y=2x + 17, even though it's not accurate, but among the options, D is the closest? Wait no, 2x+17 at x=2 is 21, but table has 19. 2x+15: x=2,19 (yes!). But that's not an option. Wait, maybe the correct answer is D, as the other options are worse. Or maybe I misread the table. Wait the table: x=0,y=11; x=2,y=19; x=4,y=27; x=6,y=35. So the pattern is y = 4x + 11. But since that's not an option, and the only linear options are C and D, and C is wrong (y=-17 at x=0), D is the only linear option with y=17 at x=0, close to 11, and maybe the problem has a typo. So the answer is D? Wait no, wait 2x + 17: x=0,17; x=2,21; x=4,25; x=6,29. No, that's not matching. Wait the square root options: let's test x=2 in Option A: \( y = 11\sqrt{2 + 0.3}-4.3≈11*1.5166 - 4.3≈16.68 - 4.3≈12.38 \), no. Option B: x=2: \( 11\sqrt{2 - 0.3}+4.3≈11*1.3038 + 4.3≈14.34 + 4.3≈18.64 \), close to 19. x=4: \( 11\sqrt{4 - 0.3}+4.3≈11*1.9235 + 4.3≈21.16 + 4.3≈25.46 \), no, table has 27. Option A: x=4: \( 11\sqrt{4 + 0.3}-4.3≈11*2.0736 - 4.3≈22.81 - 4.3≈18.51 \), no. So none of the options are perfect, but the linear options: C is wrong (y=-17 at x=0), D is y=17 at x=0, close to 11, and the first difference is 8 (x+2), but D has difference 4 (x+2: 21-17=4). So maybe the problem is designed to have D as the answer, even with the discrepancy.

Answer:

D. \( y = 2x + 17 \)