which are characteristics of the graph of the function $f(x) = (x + 1)^2 + 2$? check all that apply.\n\nthe…

which are characteristics of the graph of the function $f(x) = (x + 1)^2 + 2$? check all that apply.\n\nthe domain is all real numbers.\nthe range is all real numbers greater than or equal to 1.\nthe $y$-intercept is 3.\nthe graph of the function is 1 unit up and 2 units to the left from the graph of $y = x^2$.\nthe graph has two $x$-intercepts.

which are characteristics of the graph of the function $f(x) = (x + 1)^2 + 2$? check all that apply.\n\nthe domain is all real numbers.\nthe range is all real numbers greater than or equal to 1.\nthe $y$-intercept is 3.\nthe graph of the function is 1 unit up and 2 units to the left from the graph of $y = x^2$.\nthe graph has two $x$-intercepts.

Answer

Explanation:

Step1: Identify the domain

The function $f(x) = (x + 1)^2 + 2$ is a polynomial, so its domain is all real numbers. $$D = (-\infty, \infty)$$

Step2: Determine the range

The vertex is $(-1, 2)$ and the parabola opens upward, so the range is $y \geq 2$. $$R = [2, \infty)$$

Step3: Calculate the y-intercept

Set $x = 0$ to find the y-intercept. $$f(0) = (0 + 1)^2 + 2 = 1 + 2 = 3$$

Step4: Analyze transformations

The graph is shifted 1 unit left and 2 units up from $y = x^2$. $$f(x) = (x - (-1))^2 + 2$$

Step5: Check for x-intercepts

Since the minimum value is 2, the graph never crosses the x-axis. $$(x + 1)^2 + 2 = 0 \Rightarrow (x + 1)^2 = -2 \text{ (No real solution)}$$

Answer:

The domain is all real numbers. The y-intercept is 3.