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What is the standard form of the function [tex]\( f \)[/tex]?

[tex]\[ f(x) = 4(x + 6)^2 + 5 \][/tex]

Substitute numerical values into the equation for [tex]\( a, b \)[/tex], and [tex]\( c \)[/tex].



Answer :

To rewrite the given function [tex]\( f(x) = 4(x + 6)^2 + 5 \)[/tex] in its standard form, we need to expand it and rewrite it in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex].

1. Start with the given function:
[tex]\[ f(x) = 4(x + 6)^2 + 5 \][/tex]

2. Expand [tex]\( (x + 6)^2 \)[/tex]:
[tex]\[ (x + 6)^2 = x^2 + 12x + 36 \][/tex]

3. Substitute this expansion back into the function:
[tex]\[ f(x) = 4(x^2 + 12x + 36) + 5 \][/tex]

4. Distribute the 4 across the terms inside the parentheses:
[tex]\[ f(x) = 4x^2 + 48x + 144 + 5 \][/tex]

5. Combine the constants 144 and 5:
[tex]\[ f(x) = 4x^2 + 48x + 149 \][/tex]

Therefore, the standard form of the function is:
[tex]\[ f(x) = 4x^2 + 48x + 149 \][/tex]

The values of the coefficients are:
[tex]\[ a = 4, \quad b = 48, \quad c = 149 \][/tex]