Question

How do you write $f\left( x \right)={{x}^{2}}-4x-10$ in vertex form?

Answer 1

In the above mentioned question we need to change the equation in the vertex form for which we are first going to write down the general equation of vertex and then from the coefficients of the above mentioned equation get the vertex point (h,k) and then substitute this point in the vertex equation to get the final vertex form.

Complete step-by-step solution:
In the above question that has been stated where we need to find the vertex form for the given equation for this we are going to first write the general equation of vertex which is:
$y=a{{\left( x-h \right)}^{2}}+k$
Now to write any quadratic equation in vertex form we need to find the vertex coordinates which are h and k. To find those coordinates we will first find the x coordinate of the vertex which can be found with the help of the formula where $x=-\dfrac{b}{2a}$ in this formula x is the x coordinate of the vertex a is the coefficient of the first term of the quadratic equation i.e. ${{x}^{2}}$ and b is the coefficient of the second term of the quadratic equation i.e. x, when we substitute the coefficient values in the formula we will get the x coordinate of the vertex. Now we can find the values of “a'' and “b” from the equation that has been mentioned in the question and we get it as $a=1$ and $b=-4$ by substituting this we will get the value of x coordinate of the vertex which will be:

& \Rightarrow {{x}_{vertex}}=-\dfrac{-4}{2} \\\ & \Rightarrow {{x}_{vertex}}=2 \\\ \end{aligned}$$ Now that we have got the x coordinate of the vertex we need to find the y coordinates of the vertex which can be found by substituting the value of x coordinate of the vertex in the quadratic equation mentioned in the question and we will get the y coordinate of the vertex as: $$\begin{aligned} & \Rightarrow {{y}_{vertex}}={{\left( 2 \right)}^{2}}-4\left( 2 \right)-10 \\\ & \Rightarrow {{y}_{vertex}}=-14 \\\ \end{aligned}$$ Now that we know both the coordinates of the vertex we can get the equation of vertex form. As we know that h and k are none other than the x and y coordinate of vertex point we will substitute it in the general equation of vertex form which will result us with the final vertex equation. So the final equation of the vertex is: $$\begin{aligned} & \Rightarrow y=1{{\left( x-\left( 2 \right) \right)}^{2}}+\left( -14 \right) \\\ & \Rightarrow y={{\left( x-2 \right)}^{2}}-14 \\\ \end{aligned}$$ **The vertex form of the quadratic equation formed is $$y={{\left( x-2 \right)}^{2}}-14$$.** **Note:** In this type of question there is another way to find the vertex point without using the above formula we will just have to convert this above statement into a parabolic equation which will be in the form of $$4a\left( y-b \right)={{\left( x-c \right)}^{2}}$$ in this equation c and b will become the vertex of the parabola with h as c and k as b and then we will substitute this in the general vertex formula.