Given
The function f(x) is defined as:
[tex]f(x)\text{ = \lparen x + 5\rparen}^2\text{ - 10}[/tex]
The zeros of a function are the values of x when f(x) is equal to 0.
Solving for the zeros:
[tex]\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\sqrt{10} \\ x\text{ = -5 }\pm\text{ }\sqrt{10} \\ x\text{ = -1.837 or -8.16} \end{gathered}[/tex]
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.
[tex]\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}[/tex][tex]\begin{gathered} 2x\text{ + 10 = 0} \\ 2x\text{ = -10} \\ Divide\text{ both sides by 2} \\ x\text{ = -5} \end{gathered}[/tex]
Next, we substitute the value of x into f(x):
[tex]\begin{gathered} f(-5)\text{ = \lparen-5+ 5\rparen}^2\text{ -10} \\ \text{= -10} \end{gathered}[/tex]
Hence, the vertex is (-5, -10)
Hence, the correct option is vertex is (-5, -10) (Option B)
