Polynomial Functions PDF

# Lesson #1: Polynomial Functions ## Polynomial Function A polynomial function can be written in the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where n (the highest exponent of the variable) is a nonnegative integer known as the degree and $a_i$ (the coefficients of the variable) are...

# Lesson #1: Polynomial Functions ## Polynomial Function A polynomial function can be written in the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where n (the highest exponent of the variable) is a nonnegative integer known as the degree and $a_i$ (the coefficients of the variable) are real numbers. The coefficient $a_n$ is the leading term of the polynomial function while $a_0$ (the term without a variable) is the constant. **Example:** $P(x) = x^6$ ## Standard Form of a Polynomial Function Written in descending powers of the independent variable. **Example:** The standard form of the polynomial function $P(x) = x + 7x^2 + 1x$ is $P(x) = -x^3 + 7x^2 + 1x + 9$. **Determine which of the following are polynomial functions and write them in standard form.** 1. $P(x) = x^3 - 4x^2 + 3x - 5$ 2. $f(x) = \sqrt{2 + 7x}$ 3. $P(x) = 3x - 7x^2 + 4$ 4. $x^3 + 2x^2 + 1x^0$ 5. $f(x) = 2 + \sqrt{x}$ 6. $g(x) = x^2$ |$POLYNOMIAL FUNCTION$ | $DEGREE$ | $NAME$ | |:------------------|:------------------------------------------|:---------------------| |$P(x) = 6$ | $0$ | Constant Function | |$P(x) = 6x - 9$ | $1$ | Linear Function | |$P(x) = x^2 - 5x + 1$ | $2$ | Quadratic Function | |$P(x) = -5x^3 + 9x^2 - 8$ | $3$ | Cubic Function | |$P(x) = x^4 - 2x^3 + 13x^2 + 15x$ | $4$ | Quantic Function | |$P(x) = 11x^5 + 3^2$ | $5$ | Quartic Function | **The function $f(x) = \sqrt{2 + 7x}$ is NOT a polynomial function because there is a variable under the radical sign.** # Polynomial Function ## Example: The polynomial function $P(x) = x^3 - 4x^2 + 3x - 5$ has the following: - The leading term is $x3$, the leading coefficient is 1, and the constant term is 5. ## Finding the Distance Between Two Points 1. **A(5, 3) and B(1, 1)** $D = (x_2-x_1)^2 + (y_2-y_1)^2$ $D = (1-5)^2 + (- 1-3)^2$ $D = (-4)^2 + (-4)^2$ $D = 16 + 16$ $D = 32$ $D = \sqrt{16 \cdot 2}$ $D = 4\sqrt2$ 2. **Find the distance between two points** **A(1, 2) and B(2, 5)** $D = (x_2 - x_1)^2 + (y_2 - y_1)^2$ $D = (2-1)^2 + (5-2)^2$ $D = 1^2 + 3^2$ $D = 1 + 9$ $D = 10$ $D = \sqrt{10}$ 3. **You are given the points (-8, 10) and (-6, 7)** $D = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}$ $D = \sqrt{(-6+8)^2 + (7-10)^2}$ $D = \sqrt{2^2 + (-3)^2}$ $D = \sqrt{4 + 9}$ $D = \sqrt{13}$ ## Finding a Point a Specific Distance Away from Another Point **Name a point that is $\sqrt2$ away from (-1, 5)** $D = (x_2-x_1)^2 + (y_2-y_1)^2$ $D^2 = (x_2-x_1)^2 + (y_2 - y_1)^2$ $D^2 = 2$ $(x_2-x_1)^2 + (y_2 - y_1)^2 = 2$ We are looking for any point that is the distance $\sqrt2$ from point (-1, 5). So, let's try these values: (1, 3) $D = (1 + 1)^2 + (3 - 5)^2$ $D = 2^2 + (-2)^2$ $D = 4 + 4$ $D = 8$ $D = \sqrt{8}$ $D = 2\sqrt{2}$ This is not correct, so let's try (2, 1): $D = (2+1)^2 + (1 - 5)^2$ $D = 3^2 + (-4)^2$ $D = 9 + 16$ $D = 25$ $D = \sqrt{25}$ $D = 5$ This is not correct either, so let's try (0, 6): $D = (0 + 1)^2 + (6 - 5)^2$ $D = 1^2 + 1^2$ $D = 1 + 1$ $D = 2$ $D = \sqrt{2}$ A point that is the distance $\sqrt{2}$ from (-1, 5) is (0, 6). ## Glossary of Terms ### Arc A part of a circle. ### Arc Length The length of an arc. It can be determined by using the proportion $\frac{A}{360} = \frac{l}{2\pi r}$, where $A$ is the degree measure of the arc, $r$ is the radius of the circle, and $l$ is the arc length. ### Central Angle An angle formed in a circle whose vertex is the center of the circle. ### Common External Tangents Lines that are tangent to two circles and do not intersect the segments joining the centers of the two circles. ### Common Internal Tangents Lines that intersect the segments joining the centers of the two circles. ### Tangent A line that intersects the circle at only one point called the point of tangency. ### Congruent Arcs Arcs of the same circle or congruent circles with equal measures. ### Congruent Circles Circles with congruent radii. ### Point of Tangency The point of intersection of the tangent line and circle. ### Secant A line that intersects a circle at exactly two points. A secant contains a chord of a circle. ### Semicircle An arc measuring one-half the circumference of a circle. ### Sector of a Circle The region bounded by an arc of the circle and the two radii to the endpoints of the arc. ### Segment of a Circle The region bounded by an arc and the segment joining its endpoints. ### Inscribed Angle An angle whose vertex is inside and whose sides contain chords of the circle. ### Intercepted Arc The arc of the circle that lies in the interior of the angle and has endpoints on the angle. ### Major Arc An arc of a circle whose measure is greater than that of a semicircle. ### Minor Arc An arc of a circle whose measure is less than that of a semicircle. ## Equation of the Circle **Standard Form** The equation of the circle is: $x^2 + y^2 = r^2$ Where the center is the origin (0, 0) and $r$ is the radius. **C(0, 0): Radius r** **2nd Standard Form** $(x-h)^2 + (y - k)^2 = r^2$ Where the center C is the point (h, k) and r is the radius. **C(h, k): Radius r** **Examples:** 1. $(x+1)^2 + (y+3)^2 = 4$ - Center: (-1, -3) - Radius: 2. 2. **Ends of a diameter: (1, 8) and (4, 5)** Midpoint Formula: $(x, y) = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$ $(x-11)^2 + (y + 8)^2 = 74$ ## How to Graph a Polynomial Function ### 1. **y = x^(local function)** ### 2. **If x is positive.** * **y = x^2 / x^2 + 10x (quadratic function)** ### 3. **If x is negative.** * **y = x^2 / x^2 + 10x (quadratic function)** ### 4. **If x is positive.** ### 5. **If x is negative.** ## Characteristics of a Polynomial * **Smooth** * **Continuous** * **May have sharp edges.** ## Not a Polynomial * **No sharp edges** * **May have a hole.** ## Synthetic and Quadratic Graph **1.** $x^3 - 2x^2 - 3x = 0$ $x(x+1)(x-3) = 0$ $x = 0$ $x = -1$ $x = 3$ * (0, 0) y-intercept * (-1, 0) * (3, 0) **2.** $(x-1)^3 (x+2)^2(x+3)$ $(x-1)(x+2)^2(x+3)$ * $x = 1$ * $x = -2$ * $x = -3$ $(1, 0)$ $(-2, 0)$ $(-3, 0)$ **3.** $-(x+2)^2(x-1)(x+4)^3$ * $x = -4$ * $x = -2$ * $x = 1$ $(-4, 0)$ $(-2, 0)$ $(1, 0)$ **4.** $(x+1)(x-2)(x+3)$ * $x = -1$ * $x = 2$ * $x = -3$ ## How the Graph Will Start and End * **Even Degree:** If the degree of the polynomial is even, the graph will start and end going in the same direction: * **Positive Leading Coefficient:** The graph will start and end going up. * **Negative Leading Coefficient:** The graph will start and end going down. * **Odd Degree:** If the degree of the polynomial is odd, the graph will start and end going in opposite directions: * **Positive Leading Coefficient:** The graph will start going down and end going up. * **Negative Leading Coefficient:** The graph will start going up and end going down. ## Table of Signs | x | (x + 4) | (x - 1) | (x + 1) | (x + 1)(x+4) | |:--------|:--------|:--------|:--------|:-------------| | x < -4 | - | - | - | - | | -4 < x < -1 | + | - | - | + | | -1 < x < 1 | + | - | + | - | | 1 < x < 4 | + | + | + | + | | x > 4 | + | + | + | + | **Above x-axis Below x-axis** ## Postulates and Theorems on Circles ### Postulates 1. **The Addition Postulate**. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. 2. **At a given point on a circle, only one line can be drawn that is tangent to the circle.** 3. **In a circle, or in congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.** 4. **In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord.** ### Theorems 1. **If an angle is inscribed in a circle, then its measure is one-half the measure of its intercepted arc.** 2. **If two tangents intersect in the exterior of a circle, then the two segments from the point of tangency to the points of intersection on the circle are congruent.** 3. **If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.** 4. **If two arcs are intercepted on the same arc then the measure of the intercepted arc is twice the measure of the inscribed angle** 5. **If the segment joining the endpoints of the chord is outside the circle, then the measure of the angle formed is one-half the positive difference of the intercepted arcs.** 6. **If the segment joining the endpoints of the chord is inside the circle, then the measure of the angle is one-half the sum of the measures of the intercepted arcs.** 7. **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.** 8. **If two secants intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.** 9. **If two inscribed angles intercept congruent arcs on the same circle, then the angles are congruent.** 10. **If a secant and a tangent intersect at a point on the circle, then the angle is a right angle.** 11. **If the measures of the two segments of a chord are equal, then the radius drawn to the point of tangency is perpendicular to the chord.** # Circle ## Undefined Terms of Geometry ### Point * **(DOT)** * Dimensionless * By a point * Capital letter ### Line * It has length but no width. * It extends in both directions. * Capital letter * Line segment * Ray * Line ### Plane * It has length * Has width (infinite) * It can extend in all directions * Capital letter * Surface. ## Two Subsets of a Line ### Segment * Segment AB or segment BA * AB or BA * Ray AB * Ray BA ## Angle * **Notation:** The symbol "<" followed by the letter representing the vertex followed by two letters each representing the end points. *<ABC, <CAB, <BAC> * **Types of Angles:** * *Central angle* * *Inscribed angle* * *Intercepted arc* ## Circle ### Radius (OR) * A segment whose endpoints are the center of the circle and a point on the circle. ### Chord * A segment whose end points are two points on a circle. ### Diameter * A chord that passes through the center of the circle. ### Central Angle * An angle whose vertex is the center of the circle and whose sides are the two radii of the circle.