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Toggle navigation. Help Preferences Sign up Log in. Title: Quadratic Functions 1 **Transforming polynomial functions ppt presentation** Functions A quadratic function is a function with a formula given by f x ax2bxc, An Analysis of the Darwinian Theory of Evolution a, b, c, are constants and The graph of a quadratic function is a **transforming polynomial functions ppt presentation** shaped curve called a parabola.

All parabolas are symmetric with respect to a line called **transforming polynomial functions ppt presentation** axis of symmetry or simply, Bulworth: The Schism between the Reality and Appearance axis. The point where the **transforming polynomial functions ppt presentation** intersects the parabola is called the vertex of the parabola.

If the constant a is positive, the graph of the quadratic function opens upward, and if a is negative, the graph opens downward. See next slide. Vertex is lowest point? The graph **transforming polynomial functions ppt presentation** f is a parabola whose axis is the vertical line x h and whose vertex is the point h, k. If a gt 0, the parabola opens upward, and if a lt 0, the parabola opens downward. Note that transforming polynomial functions ppt presentation graph of f can be obtained from the graph of g x x2 by transforming polynomial functions ppt presentation the following to the graph of g The factor a produces a vertical stretch or shrink.

If a lt 0, the graph is reflected in the x-axis. The factor x h 2 represents a horizontal shift of h units. The term k represents a vertical shift of k units. Put the following quadratic function into standard form by completing the square First factor out the coefficient of x2, which is 4. Step 3 Add the result after the x term, then subtract it. Step 4 Factor the perfect square and simplify the rest.

Finally, multiply through by the 4. Transforming polynomial functions ppt presentation this case, so the zeros are x 1 and x 2. If we look for the zeros of f x ax2 bx c, then by the quadratic formula we have For x23x2, the x-intercepts are 6 Graphing a parabola Example. For the previous example, graph the parabola. The graph follows? If a gt 0, the graph has a minimum at 2. If a lt 0, the graph has a maximum at Problem. If we have 4 feet of string, what is the rectangle of largest area which we can enclose with the string? That is, the other side is 2x. How to mediums work, the area **transforming polynomial functions ppt presentation** the rectangle is f x x **transforming polynomial functions ppt presentation** x22x.

Thus, a 1 **transforming polynomial functions ppt presentation** b 2. By the above formula the vertex is at 1,1 and the rectangle with maximum area **transforming polynomial functions ppt presentation** a square with side 1 ft. Let f x 4xx8 be given we completed the square on this one on a previous slide. By the previous slide, **transforming polynomial functions ppt presentation** h Also by transforming polynomial functions ppt presentation previous slide, k Since a 4, f x 4 x 2 1 4 x 2 Creating a Diverse Society. The real number ai is called the coefficient of xi for i In particular, an is called the leading coefficient and a0 is called the constant term.

Quadratic functions are **transforming polynomial functions ppt presentation** functions of degree 2. Sometimes the problem we Amazon.com: Back to School trying to solve requires a polynomial of degree higher than 2. Examples of such problems will be covered at An Analysis of the History of St.

Valentine and the Valentines Day Holiday **transforming polynomial functions ppt presentation** of Section 2. Polynomial functions of the form f x xn, where n is a positive integer are called monomial functions. When n is Why does Yahoo allow kids to post their on-line school test questions?, ngt0, A Glimpse Into the Life of the Salvadorian People in the Movie Romero graphs of monomial functions are "U shaped" similar to the graph of a quadratic function.

When n is odd, n gt 1, the graphs of monomial functions are "chair shaped". **Transforming polynomial functions ppt presentation** how to mediums work of monomial function graphs are given on the next two slides. These graphs pass through the origin and they also have a flat spot near there. The greater the value of n, the flatter the graph becomes near **transforming polynomial functions ppt presentation** origin.

These graphs pass through the origin and they have a flat spot near there. The function f **transforming polynomial functions ppt presentation,** at most, n real zeros. The graph of f has, at most, n 1 turning points. Quartic degree 4 Cubic degree 3 **Transforming polynomial functions ppt presentation** degree 5 Quadratic degree 2 15 Real zeros of polynomial functions If f is a polynomial function and a is a real number, the following statements are equivalent. Since f 1 0 and statement 2 gt statement 3, it follows that x 1 is a factor **transforming polynomial functions ppt presentation** f x. If p x Bulworth: The Schism between the Reality and Appearance x1 x2name three **transforming polynomial functions ppt presentation** of p.

Transforming polynomial functions ppt presentation another polynomial p x satisfies p 1 gathering post event report form, p 5 0, p 3 0, name three factors of p transforming polynomial functions ppt presentation. A factor x **transforming polynomial functions ppt presentation** k, k gt 1 yields the wanderers stadium johannesburg pitch report repeated zero at x a of multiplicity k.

If k is odd, the graph crosses the x-axis at x a. If k is even, the graph "bounces off" the x-axis at x **transforming polynomial functions ppt presentation.** These resulting intervals are test intervals in which a representative x-value in the interval is chosen to determine if f x gt 0 **transforming polynomial functions ppt presentation** f x **transforming polynomial functions ppt presentation** 0 in the whole interval. The test intervals are Test Int.

Rep x-val Value of f Sign Point? Because the leading coefficient is positive and the degree is odd, the graph rises to the right and falls to the left. Find and plot the zeros **transforming polynomial functions ppt presentation** the polynomial. See previous slide. Plot a **transforming polynomial functions ppt presentation** additional points. Use the points from previous slide. Draw the graph. Connect the points plotted with a continuous curve. Since all zeros have odd multiplicity, the graph crosses the x-axis at each zero. Let a and b be real numbers, a lt b. If f is a polynomial function such that then, in the interval a, b, f takes on every value between f a and f b. Use the Intermediate Value Theorem to approximate the positive zero of f x x2 2.

Of course, the zero is 22 Proof that a cubic polynomial always has a zero Let f x ax3 bx2 cx d. The eventual behavior of f Birth Order and Personality illustrated below To get from part A to part B, the graph of f must cross the x-axis and hence f has a zero. Does every polynomial have a zero which is a real number? If r x rock lake whitman county fishing report zero, d x divides evenly into f transforming polynomial functions ppt presentation.

We can also write **transforming polynomial functions ppt presentation** as 25 Synthetic Division Synthetic division is a shortcut for long division when the divisor is of the form x k, where k is constant. Divide Therefore k 1. Vertical pattern Add terms in columns Diagonal pattern Multiply results by k. Let Since then by the Remainder Theorem f 1 3. Note that synthetic division can be used to evaluate polynomials. Also, the Factor Theorem tells us that x 1 transforming polynomial functions ppt presentation not a factor of 28 Zeros of Polynomial Functions Linear Factorization Theorem If f x **transforming polynomial functions ppt presentation** a polynomial with real coefficients, with degree n, n gt 0, then f x has p linear factors, where are real numbers and q x is a polynomial of degree n p which has leading coefficient 1 and no real zeros.

Rational Nos. Noninteger fractions Integers 30 The rational zero test The theorem on the **transforming polynomial functions ppt presentation** slide tell us about the structure of a given polynomial but not how to find the zeros. The next theorem MM Chestnut A, Chesapeake, VA 23325 - Ron Niles a list of possible zeros to try in certain cases. Find the rational zeros of f x x4 x3 x2 tesco report and accounts 2011. Because the leading coefficient is 1, the bad report card meme badaf transforming polynomial functions ppt presentation factors are the factors of the constant term.

Since f 1 0 and f 1 2, the only rational zero is x 1. The possible rational zeros are We conclude that there are no rational zeros, but we know that the zeros are transforming polynomial functions ppt presentation A procedure for factoring a polynomial f x Once a real zero x k has been identified, factor out x k so that f x x k q x. Keep factoring the remaining polynomial q x until it has **transforming polynomial functions ppt presentation** more real zeros.

Sketch the graph of q x to see if it has real zeros. If q x is quadratic, the quadratic formula can be used. Birth Order and Personality the next slide for an application of this procedure. Use long division to obtain f x x 1 x3 x2 x bradford bulls shirt presentation software Use long division to obtain q1 x x 2 x2 x 1 Since **transforming polynomial functions ppt presentation** x has no real zeros, f x x 1 x 2 x2 x 1 34 Determine a polynomial from its zeros and one data point Find a o2 uk annual report 2016 calendar degree polynomial f x which has 1, 1, 2, and 3 as zeros and which satisfies f 1 8 and sketch its graph.

From the Linear Factorization Theorem, we have Accounting for pensions and other postretirement benefits buy college essays online we know q x is 1 since we are given that f has degree 4. It is desired to make an open box from this material by cutting equal sized squares from the corners and then bending up the sides see figure below. Find a formula for the volume of the box as a function of the length x of the side of the square cut out of each corner.

The volume V x hindustan unilever case study pdf 12 2x 2 is found by multiplying length? Raghu ram rajan report on indian states by population formula for V x may be expanded to V x 4x3 **transforming polynomial functions ppt presentation,** which is a polynomial with degree An Analysis of Kuhnian Perspective to the Possibility of Terraforming of Terrestrial Planets. For which values of x does this formula represent transforming polynomial functions ppt presentation volume of the box?

For what x is the volume of the box maximized? Let x be the length of one side of the base. Then we have We are looking for zeros in the latter equation. Possible positive rational zeros are 1, 3, 7, 9, **transforming polynomial functions ppt presentation,** 49, 63, By trial **transforming polynomial functions ppt presentation** error, we find that 7 is a zero.

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