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Credits: LACUANAN, GABRIEL CLARK, EVANGELISTA, MARCUS ANDREI 10 - EINSTEIN (Updated as of 01-21-2024) TOPIC OUTLINE
- Polynomial Functions
- Zeroes of Polynomial Functions
- Graphing Polynomial Functions
- Circle and Equation of a Circle
- Angles formed by Tangents and Secants
- Power Theorems POLYNOMIAL FUNCTION A polynomial function is a function that can be expressed in the form of a polynomial. The definition can be derived from the definition of a polynomial equation. It can also be defined in the form of: where an, an−1, an−2,... , a 2 , a 1 , and a 0 are real numbers, an ≠ 0, and n is a nonnegative integer. The domain of a polynomial function is the set of real numbers.Its range depends on whether n is even or odd, and on the value of the leading coefficient an. A polynomial function written with its terms arranged according to descending powers of x is said to be in standard form. The value of n in p(x) determines the degree of the polynomial. To evaluate a polynomial function means to find the value of the function at a given value of x. ZEROES OF A POLYNOMIAL FUNCTION The zeroes of polynomial function P are the roots of the corresponding polynomial equations P(x) = 0. This implies that if P c = 0, then c is a zero of P(x). Simply said, the number of zeroes in a polynomial function can only be fewer or equal to its degree, not more. GRAPHING POLYNOMIAL FUNCTIONS Graphs behave differently at various x-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off. GRAPHING POLYNOMIAL FUNCTIONS Graphs behave differently at various x-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off. If a polynomial contains a factor of the form (x − h)p, the behavior near the x-intercept is determined by the power p. We say that x = h is a zero of multiplicity p. ⇨ For zeros with even multiplicities , the graphs touch or are tangent (bounce off) to the x-axis. ⇨ For zeros with odd multiplicities , the graphs cross or intersect the x-axis. ⇨ The higher the multiplicity , the flatter the curve is at the zero. ⇨ The sum of the multiplicities is the degree of the polynomial function. SUMMARY
Credits: LACUANAN, GABRIEL CLARK, EVANGELISTA, MARCUS ANDREI 10 - EINSTEIN (Updated as of 01-21-2024) GRAPH BEHAVIORS AT X - AXIS CROSSING Happens when the degree is an odd integer. This can happen with degrees like linear or cubic. BOUNCING Happens when the degree is an even integer. This can happen with degrees like quadratic or quartic. LINE-LIKE (LINEAR) Happens when the factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. BOUNCE OFF (QUADRATIC) Happens when the factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off the horizontal axis at the intercept. S-SHAPE CROSS (CUBIC) Happens when the factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit function f(x) = x^3 EXAMPLE: Here: we graph the function: (x+3)(x-2)^2 (x+1)^3 FIRST X-INTERCEPT: The first x-intecept will be determined from (x+3). To do so:
- Equate the expression to zero. x + 3 = 0
- Transpose 3 to the other side, isolating x. x = -
Credits: LACUANAN, GABRIEL CLARK, EVANGELISTA, MARCUS ANDREI 10 - EINSTEIN (Updated as of 01-21-2024) GRAPHING Here, we are graphing: f(x) = −2(x + 3)^2 (x − 5) Step 1. The leading term, if this polynomial were multiplied out, would be − 2x^3 ,so the end behavior is that of vertically reflected cubic, with the graph falling to the right and going in opposite direction (up) on the left. Step 2. This graph has two x-intercepts. At x = − 3, the factor is squared, indicating a multiplicity of
At x = 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept. Step 3. Connect the end behavior lines with the intercepts. TURNING POINT A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). A polynomial of degree n will have, at most, n number of x-intercepts and n − 1 turning points. ⇨ The degree of a polynomial function helps us to determine the number of x-intercepts and the number of turning points. ⇨ A polynomial function of nth degree is the product of n factors, so it will have at most n
Credits: LACUANAN, GABRIEL CLARK, EVANGELISTA, MARCUS ANDREI 10 - EINSTEIN (Updated as of 01-21-2024) roots or zeros. INTERMEDIATE VALUE THEOREM If f(x) is a polynomial function with real coefficients, and f(a) and f(b) are opposite in signs, then there exists a value c between a and b such that f(c) = 0. It implies that there is a zero of f(x) between a and b. FACTORED FORM If a polynomial of lowest degree p haa horizontal intercepts at x = x 1 , x 2 ,..., xn, then the polynomial can be written in the factored form: f(x) = a(x-x 1 )p1(x-x 2 )p2…(x-xn) pn^ where the power pi on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the x-intercept.
WRITING A FORMULA GIVEN THE GRAPH OF THE
POLYNOMIAL FUNCTION
- Identify the x-intercepts of the graph to find the factors of the polynomial.
- Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor.
- Find the polynomial of least degree containing all the factors found in the previous step.
- Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor. EXAMPLE Looking at the graph of this function, it appears that there are x-intercepts at x = −3, −2, and 1. Each x-intercept corresponds to a zero of the polynomial function and each zero yields, so we can write the polynomial in factored form: h(x)=a(x+3)(x+2)(x-1) The stretch factor a can be found by using another point on the graph, like the y-intercept, (0, −6). h(x)=a(x+3)(x+2)(x-1) h(x)=(x+3)(x+2)(x-1)
Credits: LACUANAN, GABRIEL CLARK, EVANGELISTA, MARCUS ANDREI 10 - EINSTEIN (Updated as of 01-21-2024) ⇨ MINOR ARC An arc measuring less than 180 degrees or less than a circle. ⇨ SEMICIRCLE An arc measuring exactly 180 degrees or half a circle. ⇨ MAJOR ARC An arc measuring more then 180 degrees or more then a semicircle. CENTRAL ANGLE An angle with its vertex at the center of the circle. Its rays are radii. INSCRIBED ANGLES An angle with its vertex at a point on the circle. It can be thought of to be made up by two chords. SECTOR An area enclosed by two radii and the corresponding arc in the circle. SEGMENT It is the area enclosed by the chord and the corresponding arc in a circle. THEOREMS If a radius is perpendicular to a chord, then it bisects the chord. If a radius bisects a chord that is not a diameter, then it is perpendicular to the chord. The perpendicular bisector of a chord passes through the center of the circle. Congruent circles are circles that have congruent radii. Concentric circles are coplanar circles having the same center. Arc Addition Postulate: The measure of an arc formed ( intercepted ) by two adjacent and non - overlapping arcs is the sum of the measures of the arcs. If two minor arcs of a circle or of congruent circles are congruent, then the corresponding chords are congruent. If two chords of a circle are congruent or congruent circles are congruent, then the corresponding minor arcs are congruent. If two central angles of a circle or of congruent circles are congruent, then the corresponding minor arcs ore congruent. If two minor arcs of a circle or of congruent circles are congruent, then the corresponding central angles are congruent.
Credits: LACUANAN, GABRIEL CLARK, EVANGELISTA, MARCUS ANDREI 10 - EINSTEIN (Updated as of 01-21-2024) If two central angles of a circle or of congruent circles are congruent, then the corresponding chords are congruent. If two chords of a circle or of congruent circles are congruent, then the corresponding central angles are congruent. An angles inscribed in a semicircle is a right angles. If two inscribed angles intercepted the same arc or congruent arcs, then the angles are congruent. Opposite angles of an inscribed quadrilateral are supplementary. If two arcs of a circle are inscribed between parallel secants, then the arcs are congruent. If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle. If two segments from the same external point are thangent to a circle, then the two segments are congruent. Two circles are tangent to each other if and only if they are coplanar and tangent to the same line at the same point. Common External Tangent
- tangent segments that do not intersect the segment connecting the centers of the 2 circles. Common Internal Tangent - tangent segments that intersect the segment connecting the centers of the 2 circles. ANGLES FORMED BY TANGENTS AND SECANTS. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. In this example: m∠MPN = ½(Arc XY - Arc MN) If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. In this example: m∠LMG = ½(Arc CEL - Arc GL) If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive DIFFERENCE of the measures of the intercepted arcs.
