To solve this problem, let's examine the images and equations provided. Here’s a step-by-step approach:

Given Information

1. Function $f(x)$: The function $f(x)$ in blue is defined as: $f(x) = x + 2$

2. Function $g(x)$: The function $g(x)$ in red is given in the form: $g(x) = (x + 2) + k$ Here, $k$ is an unknown constant that we need to determine.

Graph Interpretation

From the graph:

• The blue line representing $f(x) = x + 2$ intersects the $y$-axis at $y = 2$.
• The red line $g(x)$ intersects the $y$-axis at a lower point than $f(x)$. To determine $k$, we can examine the vertical shift between the two lines.

Solution

Since $g(x) = (x + 2) + k$ represents a vertical shift of $f(x)$, we can calculate $k$ by measuring this vertical distance. Observing the graph:

• The $y$-intercept of $f(x) = x + 2$ is $2$.
• The $y$-intercept of $g(x)$ appears to be $-4$.

So: $k = -4 - 2 = -6$

Thus, $k = -6$.

Final Answer

The value of $k$ is -6.

Transformation Description

The function $g(x)$ is a vertical translation of $f(x)$ downward by 6 units. Therefore:

• $g(x) = (x + 2) - 6$ is obtained by shifting the graph of $f(x) = x + 2$ down by 6 units.

Would you like further details or have any questions?

Related Questions

1. What effect does changing $k$ have on the graph of $g(x)$?
2. How would the graph look if $k$ were positive?
3. What if $g(x)$ was shifted horizontally instead of vertically?
4. How would the slope of $f(x)$ and $g(x)$ affect their relative positions if they were changed?
5. Can we express the transformation in terms of a translation vector?

Tip

When examining transformations, remember that changes outside the function (like $+k$) affect the $y$-direction, while changes inside the function (like $x + h$) affect the $x$-direction.