Question

which function is a quadratic function?

\\(t(x) = (x - 4)^2 + 3\\)
\\(q(x) = (2x + 4)^3 + 3\\)
\\(r(x) = (-6x - 4)^4 + 3(x + 2)^2\\)
\\(s(x) = -(x - 6) + 3(x + 1)\\)

Explanation:

Define a quadratic function

A quadratic function is a polynomial function of degree 2, which can be written in the general form \(f(x) = ax^2 + bx + c\) where \(a
eq 0\).

Analyze the degree of each option

• For \(t(x) = (x - 4)^2 + 3\), expanding the squared term yields \(t(x) = x^2 - 8x + 16 + 3 = x^2 - 8x + 19\). The highest power of \(x\) is 2, so it is a quadratic function.
• For \(q(x) = (2x + 4)^3 + 3\), the highest power of \(x\) after expansion is 3, making it a cubic function.
• For \(r(x) = (-6x - 4)^4 + 3(x + 2)^2\), the highest power of \(x\) after expansion is 4, making it a quartic function.
• For \(s(x) = -(x - 6) + 3(x + 1)\), expanding yields \(-x + 6 + 3x + 3 = 2x + 9\). The highest power of \(x\) is 1, making it a linear function.

Identify the correct function

The only function with a degree of 2 is \(t(x) = (x - 4)^2 + 3\).

Answer:

• (A) \(t(x) = (x - 4)^2 + 3\) (Correct answer)
• (B) \(q(x) = (2x + 4)^3 + 3\)
• (C) \(r(x) = (-6x - 4)^4 + 3(x + 2)^2\)
• (D) \(s(x) = -(x - 6) + 3(x + 1)\)