the graph of ( f(x)=x^{2} ) has which of the following key features?\n a. a constant first difference\n b. a…

Explanation:

Step1: Analyze the function (y = x^{2})

The function (y=x^{2}) is a quadratic function. The general form of a quadratic function is (y = ax^{2}+bx + c). For (y=x^{2}), (a = 1), (b=0), (c = 0).

Step2: Find the vertex

The (x) - coordinate of the vertex of a quadratic function (y=ax^{2}+bx + c) is given by (x=-\frac{b}{2a}). Substituting (a = 1) and (b = 0), we get (x = 0). Substituting (x = 0) into (y=x^{2}), we get (y=0). So the vertex is ((0,0)).

Step3: Check for horizontal asymptote

A horizontal asymptote exists for rational functions (e.g., (y=\frac{f(x)}{g(x)}) where (\text{deg}(f(x))\leq\text{deg}(g(x)))). Since (y=x^{2}) is a polynomial function (not a rational function in the form that would have a horizontal asymptote), there is no horizontal asymptote.

Step4: Check for turning point

The first - derivative of (y=x^{2}) is (y^{\prime}=2x). Setting (y^{\prime}=0), we get (2x = 0\Rightarrow x = 0). The second - derivative (y^{\prime\prime}=2>0). So the function has a minimum (a turning point) at (x = 0), (y=0) (not at ((1,1))).

Step5: Check for constant first - difference

For a linear function (y=mx + b), the first - difference (\Delta y=m) (constant). For (y=x^{2}), if (x_1) and (x_2) are two values of (x), (y_1=x_1^{2}) and (y_2=x_2^{2}), (\Delta y=x_2^{2}-x_1^{2}=(x_2 - x_1)(x_2 + x_1)), which is not constant.

Answer:

d. A vertex at ((0,0))