To sketch the graph of the function $f(x) = x + 6$ by using transformations, we will follow these steps:

Step 1: Identify the base function

The base function here is: $f(x) = x$ This is a standard linear function, which represents a straight line passing through the origin with a slope of $1$.

Step 2: Describe the transformation

The given function is: $f(x) = x + 6$ This equation indicates a vertical shift of the base function $f(x) = x$.

• Vertical Shift: The $+6$ in the equation means that the graph of the line $f(x) = x$ is shifted upward by 6 units.

Step 3: Sketch the graph

• Start with the graph of the base function $f(x) = x$, which is a straight line that passes through the points $(0, 0)$, $(1, 1)$, and $(-1, -1)$.
• Now shift every point on this line upward by 6 units.

After the transformation:

• The point $(0, 0)$ moves to $(0, 6)$.
• The point $(1, 1)$ moves to $(1, 7)$.
• The point $(-1, -1)$ moves to $(-1, 5)$.

The new line has the same slope as the original, but it now crosses the $y$-axis at $6$.

Step 4: Draw the final graph

• The graph is a straight line with a slope of $1$ that passes through the point $(0, 6)$.
• The equation of the line remains linear, indicating that it continues indefinitely in both directions.

Would you like a visual sketch of this graph, or more details about its properties?

Relative Questions

1. What is the slope of the line in the transformed function $f(x) = x + 6$?
2. How do vertical shifts affect the $y$-intercept of a linear function?
3. What other types of transformations could apply to the function $f(x) = x$?
4. How would the graph change if the function were $f(x) = x - 6$?
5. What is the relationship between the slope of the line and the steepness of its graph?

Tip:

When shifting a graph vertically, only the $y$-coordinates of the points change, not the $x$-coordinates.