The factor is repeated, that is, the factor [latex]\left(x - 2\right)[/latex] appears twice. A polynomial is generally represented as P(x). A polynomial is called a univariate or multivariate if the number of variables is one or more, respectively. A polynomial having one variable which has the largest exponent is called a degree of the polynomial. Notice in the figure belowthat the behavior of the function at each of the x-intercepts is different. These are also referred to as the absolute maximum and absolute minimum values of the function. As [latex]x\to \infty [/latex] the function [latex]f\left(x\right)\to \mathrm{-\infty }[/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant. Math. Use the end behavior and the behavior at the intercepts to sketch a graph. The exponent on this factor is \( 3\) which is an odd number. This can be visualized by considering the boundary case when a=0, the parabola becomes a straight line. The graph looks almost linear at this point. [latex]{\left(x - 2\right)}^{2}=\left(x - 2\right)\left(x - 2\right)[/latex]. The graph will bounce at this \(x\)-intercept. The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice. Textbook solution for Precalculus 11th Edition Michael Sullivan Chapter 4.1 Problem 88AYU. Determine the end behavior by examining the leading term. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. (b) Is the leading coefficient positive or negative? Therefore the zero of\(-1\) has even multiplicity of \(2\), andthe graph will touch and turn around at this zero. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph below. The maximum number of turning points of a polynomial function is always one less than the degree of the function. A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. In addition to the end behavior, recall that we can analyze a polynomial functions local behavior. If a function is an odd function, its graph is symmetrical about the origin, that is, \(f(x)=f(x)\). See the graphs belowfor examples of graphs of polynomial functions with multiplicity 1, 2, and 3. Sometimes the graph will cross over the x-axis at an intercept. Example \(\PageIndex{12}\): Drawing Conclusions about a Polynomial Function from the Factors. Recall that if \(f\) is a polynomial function, the values of \(x\) for which \(f(x)=0\) are called zeros of \(f\). What would happen if we change the sign of the leading term of an even degree polynomial? To sketch the graph, we consider the following: Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0). In the figure below, we showthe graphs of [latex]f\left(x\right)={x}^{2},g\left(x\right)={x}^{4}[/latex], and [latex]h\left(x\right)={x}^{6}[/latex] which all have even degrees. Note: Whether the parabola is facing upwards or downwards, depends on the nature of a. A constant polynomial function whose value is zero. If a function has a global maximum at \(a\), then \(f(a){\geq}f(x)\) for all \(x\). We see that one zero occurs at [latex]x=2[/latex]. In the first example, we will identify some basic characteristics of polynomial functions. \[\begin{align*} f(0)&=a(0+3)(02)^2(05) \\ 2&=a(0+3)(02)^2(05) \\ 2&=60a \\ a&=\dfrac{1}{30} \end{align*}\]. In the standard form, the constant a represents the wideness of the parabola. Which statement describes how the graph of the given polynomial would change if the term 2x^5 is added? If a polynomial of lowest degree phas zeros at [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex],then the polynomial can be written in the factored form: [latex]f\left(x\right)=a{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}[/latex]where the powers [latex]{p}_{i}[/latex]on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor acan be determined given a value of the function other than the x-intercept. The figure belowshowsa graph that represents a polynomial function and a graph that represents a function that is not a polynomial. where D is the discriminant and is equal to (b2-4ac). Example \(\PageIndex{10}\): Find the MaximumNumber of Intercepts and Turning Points of a Polynomial. Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[/latex], so the end behavior, as seen in the following graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity. We can apply this theorem to a special case that is useful for graphing polynomial functions. The complete graph of the polynomial function [latex]f\left(x\right)=-2{\left(x+3\right)}^{2}\left(x - 5\right)[/latex] is as follows: Sketch a possible graph for [latex]f\left(x\right)=\frac{1}{4}x{\left(x - 1\right)}^{4}{\left(x+3\right)}^{3}[/latex]. The polynomial has a degree of \(n\)=10, so there are at most 10 \(x\)-intercepts and at most 9 turning points. Set a, b, c and d to zero and e (leading coefficient) to a positive value (polynomial of degree 1) and do the same exploration as in 1 above and 2 above. If a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], the behavior near the x-intercept his determined by the power p. We say that [latex]x=h[/latex] is a zero of multiplicity p. The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. Additionally, the algebra of finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes be impossible to findby hand. If the graph intercepts the axis but doesn't change sign this counts as two roots, eg: x^2+2x+1 intersects the x axis at x=-1, this counts as two intersections because x^2+2x+1= (x+1)* (x+1), which means that x=-1 satisfies the equation twice. Thus, polynomial functions approach power functions for very large values of their variables. Legal. Notice in the figure to the right illustrates that the behavior of this function at each of the \(x\)-intercepts is different. How many turning points are in the graph of the polynomial function? The graph has a zero of 5 with multiplicity 3, a zero of 1 with multiplicity 2, and a zero of 3 with multiplicity 2. \( \begin{array}{ccc} Graphing a polynomial function helps to estimate local and global extremas. Polynomial functions of degree 2 or more are smooth, continuous functions. The graph will cross the x -axis at zeros with odd multiplicities. Graph 3 has an odd degree. Find the polynomial of least degree containing all of the factors found in the previous step. To determine the stretch factor, we utilize another point on the graph. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[/latex], is an even power function, as xincreases or decreases without bound, [latex]f\left(x\right)[/latex] increases without bound. Download for free athttps://openstax.org/details/books/precalculus. We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, w. Notice that after a square is cut out from each end, it leaves a [latex]\left(14 - 2w\right)[/latex] cm by [latex]\left(20 - 2w\right)[/latex] cm rectangle for the base of the box, and the box will be wcm tall. Without graphing the function, determine the maximum number of \(x\)-intercepts and turning points for \(f(x)=3x^{10}+4x^7x^4+2x^3\). The graph of function ghas a sharp corner. If a point on the graph of a continuous function fat [latex]x=a[/latex] lies above the x-axis and another point at [latex]x=b[/latex] lies below the x-axis, there must exist a third point between [latex]x=a[/latex] and [latex]x=b[/latex] where the graph crosses the x-axis. Step 2. Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. b) This polynomial is partly factored. Recall that we call this behavior the end behavior of a function. The zero associated with this factor, \(x=2\), has multiplicity 2 because the factor \((x2)\) occurs twice. A global maximum or global minimum is the output at the highest or lowest point of the function. (The graph is said to betangent to the x- axis at 2 or to "bounce" off the \(x\)-axis at 2). If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros. A polynomial function, in general, is also stated as a polynomial or polynomial expression, defined by its degree. Call this point \((c,f(c))\).This means that we are assured there is a solution \(c\) where \(f(c)=0\). If the function is an even function, its graph is symmetrical about the \(y\)-axis, that is, \(f(x)=f(x)\). We have therefore developed some techniques for describing the general behavior of polynomial graphs. \( \begin{array}{rl} This means we will restrict the domain of this function to [latex]0

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