Question

How do you find the range of a quadratic function $ f\left( x \right) = - {x^2} + 14x - 48 $ ?

Answer 1

Hint: In the given equation, we are given a quadratic function and we have to find out the range for the quadratic function. There are various methods that can be employed to find the range of the quadratic function such as completing the square method. Range is the set of values that the given quadratic function can assume.

Complete step-by-step answer:
So, we have to find the range for the quadratic function $ f\left( x \right) = - {x^2} + 14x - 48 $ .
Now, Taking negative sign common from all the terms, we get,
 $ \Rightarrow f\left( x \right) = - \left( {{x^2} - 14x + 48} \right) $
Now, we need to complete the square so as to condense it into a whole square term.
We know the algebraic identity $ {\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right) $ . So, we will try to manipulate and formulate the terms into a whole square.
So, we get,
 $ \Rightarrow f\left( x \right) = - \left( {{x^2} - \left( 2 \right)\left( 7 \right)x + 48} \right) $
Adding and subtracting $ {7^2} $ in brackets so as to complete the whole square term.
 $ \Rightarrow f\left( x \right) = - \left( {{x^2} - \left( 2 \right)\left( 7 \right)x + {7^2} + 48 - 49} \right) $
Completing the square in the bracket, we get,
 $ \Rightarrow f\left( x \right) = - \left( {{{\left( {x - 7} \right)}^2} - 1} \right) $
Opening the bracket, we get,
 $ \Rightarrow f\left( x \right) = 1 - {\left( {x - 7} \right)^2} $
Now, we know that the range of $ {\left( {x - 7} \right)^2} $ is $ [0,\inf ) $ .
Hence, we get the range of the quadratic function $ f\left( x \right) = - {x^2} + 14x - 48 $ as $ ( - \inf ,1] $ substituting the range of $ {\left( {x - 7} \right)^2} $ in $ f\left( x \right) = 1 - {\left( {x - 7} \right)^2} $ .

Note: We should keep in mind this method as it can be used to find the range of any quadratic function like the one given in the question. Care should be taken while manipulating and formulating the quadratic function so as to condense it into a whole square.