Question

The function g is given by g(x) = ax^2 − b, where a and b are real numbers. If g(2) = −5 and g(−1) = 2, find the values of a and b.​

Answer 1

a = -7/3

b = -13/3

Explanation:

Making equations in terms of 'a' and 'b' :

• g(2) = -5 = a(2)² - b ⇒ 4a - b = -5 [Equation 1]
• g(-1) = 2 = a(-1)² - b ⇒ a - b = 2 [Equation 2]

Subtract : Equation 1 - Equation 2

• 4a - b - a + b = -5 - 2
• 3a = -7
• a = -7/3

Finding b :

• -7/3 - b = 2
• b = -7/3 - 6/3
• b = -13/3

Answer 2

a = 1

b = -1

Explanation:

`g(x)=ax^2-b \quad \quad \textsf{(where }\:a\:\textsf{ and }\:b\:\textsf{ are real numbers)}`

Create 2 equations with b as the subject using the given information.

Equation 1

`\begin{aligned}g(2) &=-5\\\implies a(2)^2-b &=-5\\4a-b &=5\\ b&=4a-5\end{aligned}`

Equation 2

`\begin{aligned}g(-1) &=2\\\implies a(-1)^2-b &=2\\a-b &=2\\ b &=a-2\end{aligned}`

Equate the equations and solve for a:

`\begin{aligned}b & =b\\\implies 4a-5 & = a-2\\3a & = 3\\a & = 1\\\end{aligned}`

Substitute the value of a into Equation 2 and solve for b:

`\begin{aligned}b & =a-2\\a=1 \implies b & =1-2\\b & = -1\end{aligned}`