Question

How do you write the quadratic function in vertex form given vertex \[\left( {4,5} \right)\] and point \[\left( {8, - 3} \right)\]?

Answer 1

Hint: Here we will first write the general equation in vertex form. Then we will substitute the value of vertex in the equation. We will then substitute the values of the point in the obtained equation and simplify it further to get the value of the slope. We will then back substitute the value of slope in the obtained equation to get the required answer.

Complete step-by-step answer:
The vertex form of a quadratic equation is \[y = m{\left( {x - a} \right)^2} + b\], where the slope of the equation is \[m\] and \[a,b\] is the vertex of \[x,y\] respectively.
The vertex given to us is \[\left( {4,5} \right)\].
Substituting \[a = 4\] and \[b = 5\] in the general equation in vertex form, we get
\[y = m{\left( {x - 4} \right)^2} + 5\]…..\[\left( 1 \right)\]
Now, in order to prove that the above equation passes through the point \[\left( {8, - 3} \right)\] the point should satisfy the equation.
So, substituting \[x = 8\] and \[y = - 3\] in equation \[\left( 1 \right)\], we get
\[ - 3 = m{\left( {8 - 4} \right)^2} + 5\]
Subtracting the terms in the bracket, we get
\[ \Rightarrow - 3 = m{\left( 4 \right)^2} + 5\]
Applying the exponent on the terms, we get
\[ \Rightarrow - 3 = 16m + 5\]
Taking \[m\] variable on one side, we get
\[ \Rightarrow 16m = - 3 - 5\]
Simplifying the equation, we get
\[\begin{array}{l} \Rightarrow m = \dfrac{{ - 8}}{{16}}\\ \Rightarrow m = - \dfrac{1}{2}\end{array}\]
So, we get the value of \[m = - \dfrac{1}{2}\].
Substituting value of \[m\] in equation \[\left( 1 \right)\], we get
\[y = - \dfrac{1}{2}{\left( {x - 4} \right)^2} + 5\]

Hence, the vertex form of the vertex given is \[y = - \dfrac{1}{2}{\left( {x - 4} \right)^2} + 5\].

Note:
The vertex form of a quadratic equation is \[y = m{\left( {x - a} \right)^2} + b\] where the slope of the equation is \[m\] and \[a,b\] is the vertex of \[x,y\] respectively. The vertex form is an alternative way to write the equation of a parabola. We need to keep in mind that in vertex form the value of \[a\] is subtracted and the value of \[b\] is added. The common mistake we can make in vertex form is of the sign one should always double check the positive and negative sign in this type of form and don’t get confused in what is the value of the vertex. Vertex form is very useful when we are dealing with parabola figure questions.