Polynomial and Rational Functions

Bounded values

Infinite range

Disconnected regions

Unbounded values

90 degrees

180 degrees

360 degrees

45 degrees

The area under curve on that function's graph

The secant line intersecting two points on that function's graph

The slope or rate of change at any given point on that function's graph

The maximum or minimum values on that function's graph

A horizontal shift right by 3 units and an upward shift by 4 units.

A reflection over the x-axis followed by an upward shift by 4 units.

A vertical stretch by a factor of -3 followed by an upward shift by -4 units.

A horizontal stretch by a factor of 3 and an upward shift by 4 units.

120 degrees

60 degrees

90 degrees

45 degrees

Multiply $g(t)$ by -1

Add a negative exponent to $b$

Replace $t$ with $-t$

Replace $b$ with its reciprocal $\\left( \\frac{1}{b} \\right)$

This adjustment incorrectly quantifies the area under the curve defined by the function $f(x)$.

The calculation no longer represents any conventional geometric slope or true 'rate.'

The new ratio yields squared speed assuming $\Delta y$ represents displacement over time $\Delta x$.

It transforms into estimating vertical concavity over the specified interval range.

Jump discontinuity

Point discontinuity

Infinite discontinuity

Removable discontinuity

-12

-3

-6

-9

Where the second derivative equals zero but doesn't change sign

Between regions where the second derivative changes sign from positive to negative or vice versa

Where the slope of the graph changes from increasing to decreasing without second derivative involvement

At every peak or trough where the second derivative equals zero only