Question

1
select the correct answer.
which equation describes the function modeled in this table?

x	-2	-1	0	1	2	3	4
f(x)	6	0	-2	0	6	16	30

a. y = 2x² - 2
b. y = 2(x + 1)²
c. y = 2(x - 1)²
d. y = (2x - 1)(x - 1)

Explanation:

Step1: Test Option A with x=1

Substitute \( x = 1 \) into \( y = 2x^2 - 2 \): \( y = 2(1)^2 - 2 = 0 \). Now check x=2: \( y = 2(2)^2 - 2 = 6 \). x=3: \( y = 2(3)^2 - 2 = 16 \)? Wait, \( 2(9)-2=16 \)? No, \( 18 - 2 = 16 \)? Wait, 18-2 is 16? Wait, 29=18, 18-2=16. Wait, but x=0: \( y = 2(0)^2 - 2 = -2 \), which matches. Wait, but let's check another option. Wait, maybe I made a mistake. Let's check Option C: \( y = 2(x - 1)^2 \). For x=0: \( 2(0 - 1)^2 = 2(1) = 2 \), but the table has f(0)=-2. So C is wrong. Option B: \( y = 2(x + 1)^2 \). x=0: \( 2(1)^2 = 2 \), but table has -2. So B is wrong. Option D: \( y = (2x - 1)(x - 1) \). x=0: ( -1)(-1)=1, but table has -2. So D is wrong. Wait, wait, earlier calculation for A: x=3: 2(9) -2=16, which matches. x=4: 216 -2=30, which matches. x=-1: 2(1) -2=0, matches. x=-2: 24 -2=6, matches. x=1: 21 -2=0, matches. So A works? Wait, but wait the original table: x=0, f(x)=-2. A: 20 -2=-2, correct. x=1: 21 -2=0, correct. x=2: 24 -2=6, correct. x=3: 29 -2=16, correct. x=4: 216 -2=30, correct. x=-1: 21 -2=0, correct. x=-2: 2*4 -2=6, correct. Wait, but earlier I thought maybe A is correct. Wait, but let's recheck the options. Wait, the options are A: \( y = 2x^2 - 2 \), B: \( 2(x+1)^2 \), C: \( 2(x-1)^2 \), D: \( (2x -1)(x -1) \). So when we test each:

• Option A:
• x=-2: \( 2(-2)^2 -2 = 8 -2 = 6 \) (matches)
• x=-1: \( 2(-1)^2 -2 = 2 -2 = 0 \) (matches)
• x=0: \( 2(0)^2 -2 = -2 \) (matches)
• x=1: \( 2(1)^2 -2 = 0 \) (matches)
• x=2: \( 2(2)^2 -2 = 8 -2 = 6 \) (matches)
• x=3: \( 2(3)^2 -2 = 18 -2 = 16 \) (matches)
• x=4: \( 2(4)^2 -2 = 32 -2 = 30 \) (matches)

• Option B:
• x=0: \( 2(0 +1)^2 = 2 \) (table has -2, so no)

• Option C:
• x=0: \( 2(0 -1)^2 = 2 \) (table has -2, so no)

• Option D:
• x=0: \( ( -1)(-1) = 1 \) (table has -2, so no)

So Option A is correct. Wait, but earlier I was confused, but after rechecking, A works for all x values in the table.

Step2: Confirm by testing all x values

We tested x=-2, -1, 0, 1, 2, 3, 4 in Option A and all match the f(x) values in the table. Other options fail at least one x value (e.g., x=0 for B, C, D).

Answer:

A. \( y = 2x^2 - 2 \)