Question

It f(x)= (x+2)^2 is the vertex form of a function, select the correct statement Question 1

Answer 1

Given

The function f(x) is defined as:

`f(x)\text{ = \lparen x + 5\rparen}^2\text{ - 10}`

The zeros of a function are the values of x when f(x) is equal to 0.

Solving for the zeros:

`\begin{gathered} (x\text{ + 5\rparen}^2\text{ - 10 = 0} \\ (x\text{ + 5\rparen}^2\text{ = 10} \\ Square\text{ root both sides } \\ x\text{ + 5 = }\pm√(10) \\ x\text{ = -5 }\pm\text{ }√(10) \\ x\text{ = -1.837 or -8.16} \end{gathered}`

The vertex of the parabola is the point at which the function changes direction

First we take the derivative of f(x), set it to zero and then solve for x.

`\begin{gathered} f(x)\text{ = x}^2\text{ + 10x + 25 - 10} \\ =\text{ x}^2\text{ + 10x + 15} \\ \\ f^(\prime)(x)\text{ = 2x + 10} \end{gathered}`
`\begin{gathered} 2x\text{ + 10 = 0} \\ 2x\text{ = -10} \\ Divide\text{ both sides by 2} \\ x\text{ = -5} \end{gathered}`

Next, we substitute the value of x into f(x):

`\begin{gathered} f(-5)\text{ = \lparen-5+ 5\rparen}^2\text{ -10} \\ \text{= -10} \end{gathered}`

Hence, the vertex is (-5, -10)

Hence, the correct option is vertex is (-5, -10) (Option B)