Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations.
f(x) = 

x
 + 6

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.

Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. f(x) = x + 6

Learn how to sketch the graph of the linear function f(x) = x + 6 by applying a vertical transformation to the standard linear function f(x) = x. This problem demonstrates how to shift graphs upward and is suitable for students in Grades 8-10.

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