What kind of transformation converts the graph of f(x)=6x+3 into the graph of g(x)=
–
6x+3?

The transformation that converts the graph of \( f(x) = 6x + 3 \) into the graph of \( g(x) = -6x + 3 \) is a **reflection across the x-axis**.

### Explanation:
- The function \( f(x) = 6x + 3 \) is a linear function with a slope of 6 and a y-intercept of 3.
- The function \( g(x) = -6x + 3 \) has the same y-intercept (3) but the slope has been negated to -6. This means the graph of \( g(x) \) is a mirror image of the graph of \( f(x) \), reflected across the x-axis.

In general, reflecting a function \( f(x) \) across the x-axis changes the sign of the output (y-values) for every input (x-values). Mathematically, this is represented as:
\[
g(x) = -f(x)
\]
In this case, \( g(x) = -f(x) = -(6x + 3) = -6x - 3 \), so the graph of \( f(x) = 6x + 3 \) becomes \( g(x) = -6x + 3 \), a reflection across the x-axis.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. What is the effect of reflecting a function across the y-axis?
2. How would the graph of \( f(x) = 6x + 3 \) change if we applied a vertical shift?
3. What transformation occurs if we change the slope of \( f(x) = 6x + 3 \)?
4. How would the graph change if the y-intercept is altered instead of the slope?
5. Can a reflection across the x-axis be combined with other transformations?

**Tip:** Reflections change the orientation of a graph but keep the shape intact, making them a key tool for visualizing function transformations.

What kind of transformation converts the graph of f(x)=6x+3 into the graph of g(x)=-6x+3?

Discover how the graph of f(x) = 6x + 3 is transformed into g(x) = -6x + 3 through a reflection across the x-axis. This explanation covers linear functions, transformations, and reflections, suitable for students in Grades 9-10.

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