Question

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

Answer 1

Hint: 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.