All Flashcards
Define a polynomial function.
A function of the form , where 'n' is a non-negative integer and the 'a' values are real numbers.
What is the degree of a polynomial?
The highest power of the variable in the polynomial.
Define local maximum.
A point on the graph of a function that is a peak within a specific region.
Define local minimum.
A point on the graph of a function that is a valley within a specific region.
What is a global maximum?
The absolute highest point of the entire graph of a function.
What is a global minimum?
The absolute lowest point of the entire graph of a function.
Define the zeros of a polynomial function.
The x-values where the function crosses the x-axis (where ).
What is an inflection point?
A point on a curve where the concavity changes (from concave up to concave down, or vice versa).
What is the leading coefficient?
The coefficient of the term with the highest power in a polynomial.
Define concavity.
The direction in which a curve bends. It can be concave up or concave down.
Explain the relationship between zeros and extrema.
Between two distinct real zeros of a polynomial, there must be at least one local maximum or minimum.
How does the leading coefficient affect the end behavior of an even-degree polynomial?
If positive, the function opens upwards (global minimum). If negative, the function opens downwards (global maximum).
What do inflection points tell us about the rate of change?
Inflection points indicate where the rate of change of the function is changing its direction (increasing or decreasing).
How does the degree of a polynomial relate to the number of turning points?
A polynomial of degree 'n' can have at most n-1 turning points (local maxima or minima).
Explain the significance of critical points.
Critical points (where the derivative is zero or undefined) are potential locations of local maxima, local minima, or saddle points.
Describe the relationship between a function's derivative and its increasing/decreasing intervals.
If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.
What does concavity tell us about the second derivative?
If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.
Explain the concept of end behavior for polynomial functions.
End behavior describes what happens to the function's values as x approaches positive or negative infinity, determined by the leading term.
How are zeros, extrema, and concavity related?
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.
Describe how to determine intervals of increasing and decreasing behavior.
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).
What does the graph of a polynomial with a double root at x=a look like?
The graph touches the x-axis at x=a but does not cross it.
How can you identify local extrema on a graph?
Look for points where the graph changes direction (peaks and valleys).
What does a steep slope on a polynomial graph indicate?
A large rate of change (either positive or negative).
How can you identify inflection points on a graph?
Look for points where the concavity changes (where the curve switches from bending upwards to bending downwards, or vice versa).
What does a horizontal tangent line on a polynomial graph indicate?
A critical point (where the derivative is zero), which could be a local maximum, local minimum, or saddle point.
How does the sign of the leading coefficient affect the graph's end behavior?
Positive leading coefficient: graph rises to the right. Negative leading coefficient: graph falls to the right.
What does the graph of the derivative tell you about the original function?
Where the derivative is positive, the original function is increasing. Where the derivative is negative, the original function is decreasing.
What does the graph of the second derivative tell you about the original function?
Where the second derivative is positive, the original function is concave up. Where the second derivative is negative, the original function is concave down.
How can the number of real roots be determined from the graph?
Count the number of times the graph intersects the x-axis.
What does a flat region of the graph indicate?
The rate of change is close to zero.