Final answer:
Reflecting the function f(x)=8x first across the y-axis and then across the x-axis results in the line passing through the origin with a positive slope, but it will be below the x-axis, in the third quadrant.
Step-by-step explanation:
To find out which graph is the result of reflecting f(x) = 8x across the y-axis and then across the x-axis, we need to understand how reflections affect the coordinates of points on a graph. Reflecting across the y-axis changes the sign of the x-coordinate of a point, while reflecting across the x-axis changes the sign of the y-coordinate.
For the function f(x) = 8x, initially, the line goes through (0,0) and has a slope of 8, meaning for every 1 unit increase in x, y increases by 8 units. When we reflect this across the y-axis, f(x) becomes f(-x) = 8*(-x) = -8x. This changes the slope to -8, reflecting our line leftwards. Then, reflecting across the x-axis changes our function to -f(-x), which would be -(8*(-x)) = 8x again, however, because of the negative sign applied to the function itself, all y values are inverted, so the slope appears to remain positive, but the line is now below the x-axis, rather than above it.
The new line after both reflections would pass through (0,0) like the original, but each increase of 1 on the horizontal axis will result in a decrease of 8 on the vertical axis because the line is now in the third quadrant.
In conclusion, after reflecting f(x) = 8x across the y-axis and then across the x-axis, the new line will have a positive slope and pass through the origin, but it will lie below the x-axis.