The slopes of tangent lines at a certain point on a function ( f(x) ) show how fast the function is changing right at that point. This is called the derivative, written as ( f'(x) ).
Here are some important things to know about how functions change:
Positive Slope: When ( f'(x) > 0 ), it means the function is going up in that section. For example, if ( f'(2) = 3 ), the function is rising by 3 units for every unit of ( x ) when ( x = 2 ).
Negative Slope: When ( f'(x) < 0 ), it means the function is going down. For instance, if ( f'(4) = -2 ), this means the function is falling by 2 units for every unit of ( x ) when ( x = 4 ).
Zero Slope: If ( f'(c) = 0 ), it means the function has a special point. This point might be where the function reaches a peak, a low point, or changes direction. These points can really change how the function behaves on a graph.
Knowing about these slopes helps us draw the function's graph and understand how it acts.
The slopes of tangent lines at a certain point on a function ( f(x) ) show how fast the function is changing right at that point. This is called the derivative, written as ( f'(x) ).
Here are some important things to know about how functions change:
Positive Slope: When ( f'(x) > 0 ), it means the function is going up in that section. For example, if ( f'(2) = 3 ), the function is rising by 3 units for every unit of ( x ) when ( x = 2 ).
Negative Slope: When ( f'(x) < 0 ), it means the function is going down. For instance, if ( f'(4) = -2 ), this means the function is falling by 2 units for every unit of ( x ) when ( x = 4 ).
Zero Slope: If ( f'(c) = 0 ), it means the function has a special point. This point might be where the function reaches a peak, a low point, or changes direction. These points can really change how the function behaves on a graph.
Knowing about these slopes helps us draw the function's graph and understand how it acts.