Starting with the graph of f(x)=5x, write the equation of the graph that results from reflecting f(x) about the line y =4.

y=_______?

Thank you!

y = 4-(5x-4) = 8-5x

To find the equation of the graph resulting from reflecting f(x)=5x about the line y=4, we can use the concept of reflecting a point about a line.

First, let's reflect the original graph f(x)=5x about the line y=4. To reflect a point (x, y) about the line y=4, we need to find the same distance below the line y=4 as the original point is above it.

The original graph f(x)=5x intersects the line y=4 at the point (0, 0) since when x=0, y=5(0)=0.

To reflect the point (0, 0) about the line y=4, we need to find a point D such that the distance from D to the line y=4 is equal to the distance from (0, 0) to the line y=4. In other words, we can find the coordinates of D as (0, -8) since the distance from (0, 0) to y=4 is 4 units and the distance from (0, -8) to y=4 is also 4 units.

Now, let's find the equation of the reflected graph. We know that the line of reflection, y=4, remains fixed.

The original graph f(x)=5x intersects the line y=4 at (0, 0), and from the previous step, the reflected point D intersects the line y=4 at (0, -8).

To find the equation of the reflected graph, we need to determine the slope of the reflected line. The slope of the original line f(x)=5x is 5, and the reflected line will have the same slope.

So, the reflected graph has a slope of 5 and passes through the point (0, -8). We can use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is any point on the line and m is the slope, to write the equation.

Using (0, -8) as the point and 5 as the slope the equation is:
y - (-8) = 5(x - 0)

Simplifying:
y + 8 = 5x

The equation of the graph resulting from reflecting f(x)=5x about the line y=4 is:
y + 8 = 5x

Hope this helps! Let me know if you have any other questions.