Questions: Graph the given functions, f and g, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of f. f(x) = x^2 g(x) = x^2 + 5 Use the graphing tool to graph the functions. Click to enlarge graph

Graph the given functions, f and g, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of f.

f(x) = x^2
g(x) = x^2 + 5

Use the graphing tool to graph the functions.
Click to enlarge graph

Transcript text: Graph the given functions, $f$ and $g$, in the same rectangular coordinate system. Describe how the graph of $g$ is related to the graph of $f$. \[ \begin{aligned} f(x) & =x^{2} \\ g(x) & =x^{2}+5 \end{aligned} \] Use the graphing tool to graph the functions. $\square$ Click to enlarge graph

Solution

Solution Steps

Step 1: Identify the given functions

The given functions are:

$ f(x) = x^2 $
$ g(x) = x^2 + 5 $

Step 2: Understand the transformation

The function $ g(x) = x^2 + 5 $ is a vertical shift of the function $ f(x) = x^2 $. Specifically, $ g(x) $ is obtained by shifting $ f(x) $ upward by 5 units.

Step 3: Graph the functions

Graph $ f(x) = x^2 $:
- This is a standard parabola opening upwards with its vertex at the origin (0,0).
- Plot points such as $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, and $(2, 4)$.
Graph $ g(x) = x^2 + 5 $:
- This is the same parabola as $ f(x) $ but shifted 5 units up.
- Plot points such as $(-2, 9)$, $(-1, 6)$, $(0, 5)$, $(1, 6)$, and $(2, 9)$.