Question

How do you write \[Y = - 6{(x - 2)^2} - 9\] in standard form?

Answer 1

Hint: We will do the derivation of the given term and then we will put this value into the inverse of the function. On doing some simplification we get the required answer.

Formula Used:
Any linear equation can be written as two following way:
\[1.\] Slope-intercept form of a linear equation
\[2.\] Standard form of a linear equation
Most of the time, any linear equation is written in slope-intercept form.
\[1.\] The general equation of slope-intercept form is as following:
\[y = m.x + c\] , where \[y,x\] are variables of the equation and \[c\] is the constant term.
And, \[m\] is called slope of the line
\[m\] can have positive value or negative value, or it can be any fractional value or any real number.
So, the slope tells the character of any straight line.
\[2.\] The general form of standard form is as following:
\[Ax + By = c\] .
But the above form shall follow the following points:
\[I.\] ‘ \[A\] ’ must be a positive number, it cannot be a negative number.
\[II.\] \[A,B\] and \[C\] must be integers, they cannot be fractional numbers.

Complete step by step answer:
It is given in the question that: \[Y = - 6{(x - 2)^2} - 9\] .
Now, square the terms in R.H.S, we get:
\[Y = - 6({x^2} - 2 \times x \times 2 + {2^2}) - 9\] .
Now, simplify the terms in R.H.S, we get:
\[Y = - 6({x^2} - 4x + 4) - 9\]
On multiply we get,
\[ \Rightarrow Y = - 6{x^2} + 24x - 24 - 9\]
On adding we get,
\[ \Rightarrow Y = - 6{x^2} + 24x - 33\]
Now, by doing the further simplification, we get:
\[Y + 6{x^2} - 24x = - 33.\]
So, by comparing the above standard form of equation, we can say that the coefficients of variables as well as the constant term are an integer.

The standard form of the given equation is \[Y + 6{x^2} - 24x = - 33.\] .

Note: Points to remember:
We need to simplify any linear equation to get the variables and constant terms on the different sides of the equation.
Slope of any linear equation tells us the character of the equation and the standard form of an equation helps us the number of integral solutions that exist for the equation.