Question

graph the given functions, f and g, in the same rectangular coordinate system. select integers for x, starting with -2 and ending with 2, and describe how the graph of g is related to the graph of f. f(x)=x², g(x)=x² + 2 (simplify your answers. type ordered pairs.) use the 4 - point graphing tool to graph the equation. use a solid graph for f(x) and a dashed graph for g(x). describe how the graph of g is related to the graph of f. select the correct choice below and fill in the answer box to complete your choice. a. the graph of g is the graph of f shifted up by unit(s). b. the graph of g is the graph of f shifted down by unit(s).

Explanation:

Step1: Analyze the functions

The function \( f(x) = x^2 \) and \( g(x)=x^2 + 2 \). We know that for a function \( y = f(x)+k \), if \( k>0 \), the graph of \( y = f(x) \) is shifted up by \( k \) units, if \( k < 0 \), it is shifted down by \(|k|\) units.

Step2: Determine the shift

Here, comparing \( f(x)=x^2 \) and \( g(x)=x^2 + 2 \), we can see that \( g(x)=f(x)+2 \). So the value of \( k = 2\) which is positive. So the graph of \( g(x) \) is the graph of \( f(x) \) shifted up by 2 units. Also, from the table, for \( x = 0 \), \( f(0)=0^2 = 0\) and \( g(0)=0^2+2=2\); for \( x = 1 \), \( f(1)=1^2 = 1\) and \( g(1)=1^2 + 2=3\); for \( x=2 \), \( f(2)=2^2=4\) and \( g(2)=2^2 + 2 = 6\). The \( y \)-values of \( g(x) \) are 2 more than the \( y \)-values of \( f(x) \) for the same \( x \)-values, which confirms the vertical shift up by 2 units.

Answer:

A. The graph of g is the graph of f shifted up by \(\boldsymbol{2}\) unit(s).