Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

True or False The y-coordinate of the vertex of $f\left(x\right)=-x^2+4x+5$ is $f\left(2\right)$.

Step-by-Step Solution

Answer

The given statement is true.

Step 1. Given information.

The given statement is:

The y-coordinate of the vertex of $f\left(x\right)=-x^2+4x+5$ is $f\left(2\right)$.

Step 2. Determine whether the given statement is true or false.

The vertex of $f\left(x\right)=a{x}^2+bx+c$ is $\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$.

In the given function $f\left(x\right)=-x^2+4x+5,a=-1,b=4,c=5$.

$\begin{array}{rcl}-\frac{b}{2a}& =& -\frac{4}{2\left(-1\right)}\\ & =& -\frac{4}{-2}\\ & =& 2\end{array}$

The y-coordinate of the vertex of the given function is $f\left(2\right)$.

Step 3. Conclusion.

The given statement is true.