Chapter 1.2, Problem 16GP

Solution

To write: A quadratic function in standard form.

The standard form of the function $y=2{\left(x-3\right)}^2+9$ is $f\left(x\right)=2x^2-12x+27$ .

Given data: The quadratic function form is $y=2{\left(x-3\right)}^2+9$ .

Method/Formula used:

The product of two factors $P=(x-b)(x-c)$ using FOIL method is given by

  $P=(a+b)(c+d)=\left(a\cdot c+a\cdot d+b\cdot c+b\cdot d\right)$

If $c=a,d=b$ then, $P=(a+b)(a+b)=a^2+2a\cdot b+b^2$

Distributive property over addition: $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$ .

The form of a quadratic function $y\left(x\right)=A{x}^2+Bx+C$ is called the standard form.

Calculation:

The form of quadratic function $y=2{\left(x-3\right)}^2+9$ can be written in standard form as,

  $\begin{array}{c}f\left(x\right)=2\left[{\left(x-3\right)}^2\right]+9\leftarrow \text{Original}\text{form}\\ =2\left[x\cdot x+2\cdot x\cdot \left(-3\right)+{\left(-3\right)}^2\right]+9\leftarrow \text{Use square}\text{formula}\\ =2\left[x^2-6x+9\right]+9\leftarrow \text{Simplifed}\\ =2\left[x^2-6x+9\right]+9\leftarrow \text{Use}\text{distributive}\text{property}\\ =\left[\left(2\right)\cdot x^2+\left(2\right)\cdot \left(-6x\right)+\left(2\right)\cdot \left(9\right)\right]+9\\ =2x^2-12x+18+9\\ =2x^2-12x+27\underset{¯}{\text{Standard}\text{form}}\end{array}$

Thus, the standard form of the function $y=2{\left(x-3\right)}^2+9$ is $f\left(x\right)=2x^2-12x+27$ .