3. Answer: (D) 6x² + 4x - 5
Step-by-step explanation:
Use the standard form of a quadratic equation y = ax² + bx + c and input the given coordinates to create three equations. Then solve the system of equations.
(0, -5) → -5 = a(0)² + b(0) + c
-5 = 0 + 0 + c
-5 = c
(-1, -3) → -3 = a(-1)² + b(-1) + (-5)
-3 = a - b - 5
2 = a - b
(1, 5) → 5 = a(1)² + b(1) + (-5)
5 = a + b - 5
10 = a + b
2 = a - b
+ 10 = a + b
12 = 2a
÷2 ÷2
6 = a
Input a = 6 into either equation to solve for b:
10 = a + b
10 = 6 + b
4 = b
Input a = 6, b = 4, and c = -5 into the standard form of a quadratic equation:
[tex]\boxed{y=6x^2+4x-5}[/tex]
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4. Answer: (D) -2x² - 2x + 4
Step-by-step explanation:
Use the standard form of a quadratic equation y = ax² + bx + c and input the given coordinates to create three equations. Then solve the system of equations.
(0, 4) → 4 = a(0)² + b(0) + c
4 = 0 + 0 + c
4 = c
(-3, -8) → -8 = a(-3)² + b(-3) + (4)
-8 = 9a - 3b + 4
-12 = 9a - 3b
(2, -8) → -8 = a(2)² + b(2) + (4)
-8 = 4a + 2b + 4
-12 = 4a + 2b
-12 = 9a - 3b → 2(-12 = 9a - 3b) → -24 = 18a - 6b
-12 = 4a + 2b → 3(-12 = 4a + 2b) → -36 = 12a + 6b
-60 = 30a
÷30 ÷30
-2 = a
Input a = -2 into either equation to solve for b:
-12 = 4a + 2b
-12 = 4(-2) + 2b
-12 = -8 + 2b
-4 = 2b
-2 = b
Input a = -2, b = -2, and c = 4 into the standard form of a quadratic equation:
[tex]\boxed{y=-2x^2-2x+4}[/tex]
