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.