A function of the form $p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where 'n' is a non-negative integer and the 'a' values are real numbers.

The highest power of the variable in the polynomial.

A point on the graph of a function that is a peak within a specific region.

A point on the graph of a function that is a valley within a specific region.

The absolute highest point of the entire graph of a function.

The absolute lowest point of the entire graph of a function.

The x-values where the function crosses the x-axis (where $f(x) = 0$).

A point on a curve where the concavity changes (from concave up to concave down, or vice versa).

The coefficient of the term with the highest power in a polynomial.

The direction in which a curve bends. It can be concave up or concave down.

Between two distinct real zeros of a polynomial, there must be at least one local maximum or minimum.

If positive, the function opens upwards (global minimum). If negative, the function opens downwards (global maximum).

Inflection points indicate where the rate of change of the function is changing its direction (increasing or decreasing).

A polynomial of degree 'n' can have at most n-1 turning points (local maxima or minima).

Critical points (where the derivative is zero or undefined) are potential locations of local maxima, local minima, or saddle points.

If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

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

End behavior describes what happens to the function's values as x approaches positive or negative infinity, determined by the leading term.

Zeros are where the function crosses the x-axis. Extrema are local max/min. Concavity describes the curve's bend. They all help define the shape of the polynomial.

Find critical points by taking the derivative and setting it to zero, then test intervals between these points to see if the derivative is positive (increasing) or negative (decreasing).

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.