function analysis unit test\ngraph and compare the shapes and key features of the functions $f(x)=(x + 2)^2$…

function analysis unit test\ngraph and compare the shapes and key features of the functions $f(x)=(x + 2)^2$ and $g(x)=2x + 4$. in 3 - 5 sentences, explain how the two functions are similar and how they are different. (4 points)

function analysis unit test\ngraph and compare the shapes and key features of the functions $f(x)=(x + 2)^2$ and $g(x)=2x + 4$. in 3 - 5 sentences, explain how the two functions are similar and how they are different. (4 points)

Answer

Explanation:

Step1: Expand (f(x))

Expand (f(x)=(x + 2)^2=x^{2}+4x + 4). It is a quadratic function, and its graph is a parabola. The general form of a quadratic function is (y = ax^{2}+bx + c) ((a\neq0)), here (a = 1), (b = 4), (c = 4). The vertex of the parabola (y=ax^{2}+bx + c) is at (x=-\frac{b}{2a}), so for (f(x)), the vertex is at (x = - 2), and (f(-2)=0).

Step2: Analyze (g(x))

(g(x)=2x + 4) is a linear - function. The general form of a linear function is (y=mx + b), where (m = 2) is the slope and (b = 4) is the y - intercept.

Step3: Compare similarities and differences

Similarities: Both functions are polynomial functions. When (x=-2), (f(-2)=( - 2 + 2)^2=0) and (g(-2)=2\times(-2)+4 = 0), so they both pass through the point ((-2,0)). Differences: The shape of the graph of (f(x)) is a parabola, while the graph of (g(x)) is a straight line. (f(x)) is a non - linear (quadratic) function with a degree of 2, and (g(x)) is a linear function with a degree of 1.

Answer:

Similarities: Both are polynomial functions and pass through ((-2,0)). Differences: (f(x)) is a parabola (quadratic), (g(x)) is a straight - line (linear); (f(x)) has degree 2, (g(x)) has degree 1.