All Flashcards
What does the graph of a polynomial touching the x-axis at x=a tell you?
x=a is a zero with even multiplicity. The function's sign does not change at x=a.
What does the graph of a polynomial crossing the x-axis at x=a tell you?
x=a is a zero with odd multiplicity. The function's sign changes at x=a.
How can you identify the degree of a polynomial from its graph?
The degree is related to the number of turning points (local maxima and minima). A polynomial of degree n can have at most n-1 turning points. Also, consider the end behavior.
How can you identify real zeros from a polynomial's graph?
Real zeros are the x-intercepts of the graph.
What does the end behavior of a polynomial graph indicate?
The end behavior indicates the sign of the leading coefficient and whether the degree is even or odd.
How does an even function's graph look?
Symmetric about the y-axis.
How does an odd function's graph look?
Rotationally symmetric about the origin.
What does a flat region on a polynomial graph suggest?
It suggests a zero with a higher multiplicity or a turning point.
How can you determine the sign of a polynomial in different intervals from its graph?
Look at whether the graph is above (positive) or below (negative) the x-axis in each interval.
What does the y-intercept of a polynomial graph represent?
The value of the polynomial when x = 0, i.e., p(0).
What are the differences between real and imaginary numbers?
Real Numbers: Can be plotted on a number line, no imaginary component | Imaginary Numbers: Involve the imaginary unit i = sqrt(-1), cannot be plotted on a standard number line
What are the differences between even and odd functions?
Even Functions: Symmetric about the y-axis, f(-x) = f(x) | Odd Functions: Symmetric about the origin, f(-x) = -f(x)
What are the differences between zeros with even and odd multiplicities?
Even Multiplicity: Graph touches the x-axis and turns around, sign of f(x) does not change | Odd Multiplicity: Graph crosses the x-axis, sign of f(x) changes
Compare and contrast x-intercepts and y-intercepts.
X-intercepts: Points where the graph crosses the x-axis, y=0, represent real zeros | Y-intercepts: Point where the graph crosses the y-axis, x=0, represents the value of the function at x=0
Compare and contrast polynomials with real coefficients and polynomials with complex coefficients.
Real Coefficients: Complex roots occur in conjugate pairs | Complex Coefficients: Complex roots do not necessarily occur in conjugate pairs
What is the difference between a root and an x-intercept?
Root: A solution to the polynomial equation p(x) = 0, can be real or complex | X-intercept: A point where the graph crosses the x-axis, only represents real roots
Compare and contrast the graphs of even and odd degree polynomials.
Even Degree: Ends go in the same direction (both up or both down) | Odd Degree: Ends go in opposite directions (one up, one down)
Compare and contrast linear and quadratic functions.
Linear: Degree 1, graph is a straight line | Quadratic: Degree 2, graph is a parabola
Compare and contrast rational and irrational numbers.
Rational: Can be expressed as a fraction p/q where p and q are integers | Irrational: Cannot be expressed as a fraction, decimal representation is non-repeating and non-terminating
Compare and contrast multiplicity 1 and multiplicity 2.
Multiplicity 1: The graph crosses the x-axis | Multiplicity 2: The graph touches the x-axis and turns around.
Explain the relationship between real zeros and x-intercepts.
Real zeros of a polynomial p(x) correspond to the x-intercepts of its graph, where the graph crosses the x-axis at the point (a, 0).
Explain the significance of even multiplicity.
If a zero a has an even multiplicity, the graph of p(x) touches the x-axis at x = a but does not cross it. The function's sign does not change at this zero.
How do you determine the degree of a polynomial using successive differences?
Calculate first differences, then second, third, and so on, until you find a constant difference. The number of times you need to take differences will be the degree of the polynomial.
What is the key property of even functions?
Even functions are symmetric about the y-axis and satisfy the property f(-x) = f(x).
What is the key property of odd functions?
Odd functions are rotationally symmetric about the origin and satisfy the property f(-x) = -f(x).
Describe the behavior of a graph at a zero with odd multiplicity.
The graph crosses the x-axis at the zero. The sign of the function changes at this point.
Explain the conjugate pairs theorem.
If a polynomial has real coefficients, complex zeros always come in conjugate pairs. If a + bi is a zero, then a - bi is also a zero.
What does it mean for a function to be symmetric about the y-axis?
The function is even, meaning f(-x) = f(x). The graph is a mirror image across the y-axis.
What does it mean for a function to be symmetric about the origin?
The function is odd, meaning f(-x) = -f(x). The graph looks the same when rotated 180 degrees around the origin.
How does multiplicity affect the graph's behavior at an x-intercept?
Odd multiplicity: graph crosses the x-axis. Even multiplicity: graph touches the x-axis and turns around.