Question

write the function in standard form.
$f(x) = -5(x - 7)^2 + 13$
$f(x) = \square$
(simplify your answer. use integers or decimals for any numbers in the expression.)

Explanation:

Step1: Expand \((x - 7)^2\)

We know that \((a - b)^2=a^2 - 2ab + b^2\). Here \(a = x\) and \(b=7\), so \((x - 7)^2=x^2-14x + 49\).
The function \(f(x)=- 5(x - 7)^2+13\) becomes \(f(x)=-5(x^2-14x + 49)+13\).

Step2: Distribute the -5

Using the distributive property \(a(b + c + d)=ab+ac + ad\), we have:
\(-5\times x^2=-5x^2\), \(-5\times(-14x) = 70x\), \(-5\times49=-245\)
So \(f(x)=-5x^2 + 70x-245 + 13\).

Step3: Combine like terms

Combine the constant terms \(-245\) and \(13\). \(-245+13=-232\)
So \(f(x)=-5x^2+70x - 232\).

Answer:

\(f(x)=-5x^2 + 70x-232\)