Local extrema: peaks and valleys in a specific region. Global extrema: absolute highest and lowest points of the entire graph.

Zeros: x-values where $f(x) = 0$. Critical points: x-values where $f'(x) = 0$ or $f'(x)$ is undefined.

Even: ends point in the same direction. Odd: ends point in opposite directions.

Root: where the graph intersects the x-axis. Turning point: local max or min.

Increasing: the function's value goes up as x increases. Decreasing: the function's value goes down as x increases.

Concavity: describes the curve's bend (up or down). Slope: describes the steepness and direction of the line tangent to the curve.

Zero: point where the graph crosses the x-axis. Inflection point: point where the concavity changes.

First derivative: gives the rate of change of the function. Second derivative: gives the rate of change of the first derivative (concavity).

Local maximum: the highest point in a specific region. Global maximum: the highest point on the entire graph.

Critical point: point where the derivative is zero or undefined. Endpoint: a point at the boundary of the function's domain.

1. Set $p(x) = 0$. 2. Factor the polynomial. 3. Solve for x.

1. Find the derivative $p'(x)$. 2. Set $p'(x) = 0$. 3. Solve for x.

1. Find critical points. 2. Create a number line with critical points. 3. Test values in each interval in $p'(x)$.

1. Find critical points. 2. Use the first or second derivative test to determine if each point is a local max, min, or neither.

1. Find zeros. 2. Find extrema. 3. Determine end behavior. 4. Plot these points and sketch the curve.

1. Identify the leading term ($a_nx^n$). 2. If n is even and $a_n > 0$, both ends go to $+\infty$. 3. If n is even and $a_n < 0$, both ends go to $-\infty$. 4. If n is odd and $a_n > 0$, left goes to $-\infty$, right goes to $+\infty$. 5. If n is odd and $a_n < 0$, left goes to $+\infty$, right goes to $-\infty$.

1. Find the second derivative $p''(x)$. 2. Set $p''(x) = 0$ and solve for x. 3. Check that the concavity changes at these points.

1. Find the second derivative $p''(x)$. 2. Determine intervals where $p''(x) > 0$ (concave up) and $p''(x) < 0$ (concave down).

1. Set $p(x) = y$. 2. Solve for x.

1. Check the end behavior. 2. If the end behavior approaches infinity, there is no global max/min. 3. If the end behavior approaches a finite value, check for local max/min and compare.

The graph touches the x-axis at x=a but does not cross it.

Look for points where the graph changes direction (peaks and valleys).

A large rate of change (either positive or negative).

Look for points where the concavity changes (where the curve switches from bending upwards to bending downwards, or vice versa).

A critical point (where the derivative is zero), which could be a local maximum, local minimum, or saddle point.

Positive leading coefficient: graph rises to the right. Negative leading coefficient: graph falls to the right.

Where the derivative is positive, the original function is increasing. Where the derivative is negative, the original function is decreasing.

Where the second derivative is positive, the original function is concave up. Where the second derivative is negative, the original function is concave down.

Count the number of times the graph intersects the x-axis.

The rate of change is close to zero.