What is a Slope?
Every point of a function has a slope, defined as the ratio of the change of
f(x)
to the change of
x
.
For example,
f(x)=2x
. The slope is built into the function. For all values of
x
,
f(x)
changes twice as fast as
x
. The slope is always
2
.
In a graph, slope is visualized as the steepness of the line.
Hover over the graph. The yellow line is
y=mx
, where
m
is equal
to the height of the cursor.
The red line graphs the slope of the yellow line.
When the yellow line points
upwards, the slope is positive.
When the yellow line points downward, the slope is
negative.
When the yellow line becomes more steep, the slope increases in magnitude.
When the yellow line flattens, the slope decreases in magnitude.
For this one,
y
decreases at twice the rate by which
x
increases. If
x
goes up
1
,
y
goes down
2
. If
x
goes up
2
,
y
goes down
4
.
Thus, the red line, representing the slope of the yellow line, is always equal to
-2
.
A linear function is one with the same slope for all values of
x
.
The next function is non-linear. The slope changes.
Hover the cursor over the graph to see
a visualization of the slope.
The line is steep at the ends, where
y
changes a lot more quickly than
x
. At
x=0
, the line is flat. Thus, the red line, graphing the slope, is at
0
.
Here is an example of a non-linear function with a linear slope.
Thus, the slope changes at a constant rate for all values of
x
, but the function does not.
The previous graph was
y=x^2
. This one is
y=mx^2
. Move the cursor up-and-down to change
m
.
This example may look confusing at first.
Hover over the graph to confirm the red line matches the slope of the function.
When the yellow line is flat, the red line is
0
.
When the yellow line is increasing, the red line is positive.
When the yellow line is decreasing, the red line is negative.
For the following graphs, the slope line has been removed.
On a piece of paper, draw the function. Then, draw the slope line.
Afterwards, hover over the graph to check your answer. The white dot will trace the slope line.
f(x)=e^x
This function is special. We will see why when checking the answer.