Question

How do you find the vertex of $y=-{{x}^{2}}+18x-75$?

Answer 1

In order to find the solution of the given question, that is to find the vertex of $y=-{{x}^{2}}+18x-75$ rewrite the given equation into the vertex form that is first compare the given equation with standard quadratic form $a{{x}^{2}}+bx+c$ and then convert it into parabola vertex form $a{{\left( x+d \right)}^{2}}+e$. Then find the equation in the form of $y=a{{\left( x-h \right)}^{2}}+k$ where $\left( h,k \right)$ is the required vertex.

Complete step-by-step solution:
According to the question, given equation in the question is as follows:
$y=-{{x}^{2}}+18x-75$
Now rewrite the above equation in vertex form.
Complete the square for $-{{x}^{2}}+18x-75$
Use the form $a{{x}^{2}}+bx+c$, to find the values of $a,b$, and $c$.
$a=-1,b=18,c=-75$
Consider the vertex form of a parabola.
$a{{\left( x+d \right)}^{2}}+e$
Substitute the values of $a$ and $b$ into the formula $d=\dfrac{b}{2a}$.
$\Rightarrow d=\dfrac{18}{2\left( -1 \right)}$
Simplify the right side of the above equation, we will have:
Cancel the common factor of $18$ and $2$ by following these steps:
Factor $2$ out of $18$.
$\Rightarrow d=\dfrac{2\cdot 9}{2\cdot -1}$
Move the negative one from the denominator of $\dfrac{9}{-1}$ from the above equation, we will have:
$\Rightarrow d=-1\cdot 9$
Now multiply $-1$ by $9$.
$\Rightarrow d=-9$
Now we will find the value of $e$ using the formula $e=c-\dfrac{{{b}^{2}}}{4a}$ by following these steps:.
Simplify each term and raise $18$ to the power of $2$, we will have:
$\Rightarrow e=-75-\dfrac{324}{4\cdot -1}$
Multiply $4$ by $-1$, we will get:
$\Rightarrow e=-75-\dfrac{324}{-4}$
Divide $324$ by $-4$, we will have:
$\Rightarrow e=-75+\left( -1 \right)\cdot \left( -81 \right)$
After this multiply $-1$ by $-81$.
$\Rightarrow e=-75+81$
Now add $-75$and $81$.
$\Rightarrow e=6$
Substitute the values of $a$, $d$, and $e$ into the vertex form $a{{\left( x+d \right)}^{2}}+e$, we will have:
$-{{\left( x-9 \right)}^{2}}+6$
After this set $y$ equal to the new right side as mentioned above, we will get:
$y=-{{\left( x-9 \right)}^{2}}+6$
Use the vertex form, $y=a{{\left( x-h \right)}^{2}}+k$ to determine the values of $a,h$, and $k$.
Clearly, we can see that $a=-1,h=9\text{ }\\!\\!\And\\!\\!\text{ }k=6$.
We know that the vertex $\left( h,k \right)$ which is equal to $\left( 9,6 \right)$.
Therefore, the vertex of the given equation $y=-{{x}^{2}}+18x-75$ is $\left( 9,6 \right)$.

Note: Students can go wrong by applying the wrong vertex formula like $x=a{{\left( y-h \right)}^{2}}+k$ which is completely wrong and leads to the wrong answer. It’s important to remember that vertex formula is $y=a{{\left( x-h \right)}^{2}}+k$ where $\left( h,k \right)$ is the vertex of the given equation.