Question

How do you find the vertex? \[y = x\left( {x - 2} \right) \]

Answer 1

Hint: We need to know the basic point form of the vertex. Also, we need to know how to convert the given equation into quadratic form by expanding the terms. Vertex is also mentioned as \[\left( {h,k} \right) \] . We need to know the formula for finding the value of \[h \] . We need to know how to substitute the suitable value in the general equation instead of the variable.

Complete step-by-step solution:
The given equation is shown below,
 \[y = x\left( {x - 2} \right) \]
The above equation can also be written as,
 \[y = {x^2} - 2x \to equation\left( 1 \right) \]
We know that,
The basic form of a quadratic equation is, \[a{x^2} + bx + c = y \to equation\left( 2 \right) \]
By comparing the equation \[\left( 1 \right) \] and \[\left( 2 \right) \] , we get the value of \[a,b \] and \[c \]
 \[equation\left( 1 \right) \to y = {x^2} - 2x \]
 \[equation\left( 2 \right) \to a{x^2} + bx + c = y \]
So, we get
 \[
  a = 1 \\
  b = - 2 \\
  c = 0 \\
  \]
We know that,
Vertex can be mentioned in the form of \[\left( {h,k} \right) \] .
Let’s find \[h \]
The formula for finding \[h \] is given below,
 \[h = \dfrac{{ - b}}{{2a}} \to equation\left( 3 \right) \]
By substituting the values \[a = 1 \] and \[b = - 2 \] in the equation \[\left( 3 \right) \] , we get
 \[equation\left( 3 \right) \to h = \dfrac{{ - b}}{{2a}} \]
 \[
  h = \dfrac{{ - \left( { - 2} \right)}}{{2 \times 1}} = \dfrac{2}{2} \\
  h = 1 \\
  \]
For finding the value of \[k \] , we substitute the value o f \[h \] in the equation \[\left( 1 \right) \] instead of \[x \]
 \[equation\left( 1 \right) \to y = {x^2} - 2x \]
So, we get
 \[y = {h^2} - 2h \]
 \[
  y = {\left( 1 \right)^2} - \left( {2 \times 1} \right) \\
  y = 1 - 2 \\
  y = - 1 \\
  \]
Next, take \[y \] as \[k \] .
 So, we get \[\left( {h,k} \right) = \left( {1, - 1} \right) \] .
So, the final answer is,
 \[Vertex = \left( {h,k} \right) = \left( {1, - 1} \right) \]

Note: This question describes the operation of addition/ subtraction/ multiplication/ division. Remember the formula to find the value of \[h \] . Also, note that after finding the \[h \] value, we would consider \[h \] as \[x \] and \[y \] as \[k \] . Note that the vertex can be mentioned as \[\left( {h,k} \right) \] . Remember the basic form of a quadratic equation to find the values of \[a,b \] and \[c \] , by using these values we can easily find the answer of \[h \] . The final answer would be in the form of a point.