WebThe graph of a polynomial function will touch the x-axis at zeros with even Multiplicity (mathematics) - Wikipedia. [latex]\begin{array}{l}f\left(0\right)=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right)\hfill \\ \text{ }-2=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right)\hfill \\ \text{ }-2=-60a\hfill \\ \text{ }a=\frac{1}{30}\hfill \end{array}[/latex]. Since the graph bounces off the x-axis, -5 has a multiplicity of 2. The graphs below show the general shapes of several polynomial functions. Process for Graphing a Polynomial Determine all the zeroes of the polynomial and their multiplicity. If the polynomial function is not given in factored form: Set each factor equal to zero and solve to find the x-intercepts. If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity. helped me to continue my class without quitting job. Some of our partners may process your data as a part of their legitimate business interest without asking for consent. Web0. I was already a teacher by profession and I was searching for some B.Ed. 4) Explain how the factored form of the polynomial helps us in graphing it. We know that the multiplicity is 3 and that the sum of the multiplicities must be 6. Example \(\PageIndex{10}\): Writing a Formula for a Polynomial Function from the Graph. The degree of the polynomial will be no less than one more than the number of bumps, but the degree might be We could now sketch the graph but to get better accuracy, we can simply plug in a few values for x and calculate the values of y.xy-2-283-34-7. 3) What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph? Show more Show This graph has three x-intercepts: x= 3, 2, and 5. And so on. Reminder: The real zeros of a polynomial correspond to the x-intercepts of the graph. Typically, an easy point to find from a graph is the y-intercept, which we already discovered was the point (0. A local maximum or local minimum at \(x=a\) (sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around \(x=a\).If a function has a local maximum at \(a\), then \(f(a){\geq}f(x)\)for all \(x\) in an open interval around \(x=a\). How Degree and Leading Coefficient Calculator Works? If a zero has odd multiplicity greater than one, the graph crosses the x -axis like a cubic. Degree Figure \(\PageIndex{18}\) shows that there is a zero between \(a\) and \(b\). Consider a polynomial function fwhose graph is smooth and continuous. End behavior of polynomials (article) | Khan Academy Also, since [latex]f\left(3\right)[/latex] is negative and [latex]f\left(4\right)[/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4. \[\begin{align} x^2&=0 & & & (x^21)&=0 & & & (x^22)&=0 \\ x^2&=0 & &\text{ or } & x^2&=1 & &\text{ or } & x^2&=2 \\ x&=0 &&& x&={\pm}1 &&& x&={\pm}\sqrt{2} \end{align}\] . The Factor Theorem helps us tremendously when working with polynomials if we know a zero of the function, we can find a factor. At each x-intercept, the graph goes straight through the x-axis. The graph will bounce off thex-intercept at this value. WebCalculating the degree of a polynomial with symbolic coefficients. \\ x^2(x^43x^2+2)&=0 & &\text{Factor the trinomial, which is in quadratic form.} Figure \(\PageIndex{18}\): Using the Intermediate Value Theorem to show there exists a zero. WebThe degree of a polynomial is the highest exponential power of the variable. The polynomial function is of degree n which is 6. How to determine the degree of a polynomial graph | Math Index Solve Now 3.4: Graphs of Polynomial Functions A global maximum or global minimum is the output at the highest or lowest point of the function. WebGraphs of Polynomial Functions The graph of P (x) depends upon its degree. Example 3: Find the degree of the polynomial function f(y) = 16y 5 + 5y 4 2y 7 + y 2. Figure \(\PageIndex{6}\): Graph of \(h(x)\). \[\begin{align} x^35x^2x+5&=0 &\text{Factor by grouping.} We see that one zero occurs at [latex]x=2[/latex]. If the graph touches the x -axis and bounces off of the axis, it is a zero with even multiplicity. A polynomial p(x) of degree 4 has single zeros at -7, -3, 4, and 8. How to determine the degree and leading coefficient WebYou can see from these graphs that, for degree n, the graph will have, at most, n 1 bumps. The degree of a function determines the most number of solutions that function could have and the most number often times a function will cross, This happens at x=4. First, notice that we have 5 points that are given so we can uniquely determine a 4th degree polynomial from these points. What if our polynomial has terms with two or more variables? WebFor example, consider this graph of the polynomial function f f. Notice that as you move to the right on the x x -axis, the graph of f f goes up. Looking at the graph of this function, as shown in Figure \(\PageIndex{6}\), it appears that there are x-intercepts at \(x=3,2, \text{ and }1\). WebAs the given polynomial is: 6X3 + 17X + 8 = 0 The degree of this expression is 3 as it is the highest among all contained in the algebraic sentence given. WebSimplifying Polynomials. The graph of a polynomial will cross the x-axis at a zero with odd multiplicity. Imagine zooming into each x-intercept. For zeros with even multiplicities, the graphstouch or are tangent to the x-axis at these x-values. 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. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The end behavior of a function describes what the graph is doing as x approaches or -. Determine the degree of the polynomial (gives the most zeros possible). About the author:Jean-Marie Gard is an independent math teacher and tutor based in Massachusetts. If a reduced polynomial is of degree 3 or greater, repeat steps a -c of finding zeros. Also, since \(f(3)\) is negative and \(f(4)\) is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4. If a function has a local minimum at a, then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all xin an open interval around x= a. Suppose were given the function and we want to draw the graph. At \(x=2\), the graph bounces at the intercept, suggesting the corresponding factor of the polynomial could be second degree (quadratic). (Also, any value \(x=a\) that is a zero of a polynomial function yields a factor of the polynomial, of the form \(x-a)\).(. The same is true for very small inputs, say 100 or 1,000. WebPolynomial factors and graphs. Lets look at another type of problem. For higher odd powers, such as 5, 7, and 9, the graph will still cross through the x-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis. Similarly, since -9 and 4 are also zeros, (x + 9) and (x 4) are also factors. Lets look at another problem. In addition to the end behavior, recall that we can analyze a polynomial functions local behavior. Zeros of polynomials & their graphs (video) | Khan Academy . 3.4: Graphs of Polynomial Functions - Mathematics LibreTexts The polynomial function must include all of the factors without any additional unique binomial WebRead on for some helpful advice on How to find the degree of a polynomial from a graph easily and effectively. To find out more about why you should hire a math tutor, just click on the "Read More" button at the right! Multiplicity Calculator + Online Solver With Free Steps WebAll polynomials with even degrees will have a the same end behavior as x approaches - and . Other times, the graph will touch the horizontal axis and bounce off. Lets look at an example. These questions, along with many others, can be answered by examining the graph of the polynomial function. The higher the multiplicity, the flatter the curve is at the zero. Even though the function isnt linear, if you zoom into one of the intercepts, the graph will look linear. The Intermediate Value Theorem states that for two numbers \(a\) and \(b\) in the domain of \(f\), if \(aPolynomial Graphing: Degrees, Turnings, and "Bumps" | Purplemath Example \(\PageIndex{5}\): Finding the x-Intercepts of a Polynomial Function Using a Graph. Grade 10 and 12 level courses are offered by NIOS, Indian National Education Board established in 1989 by the Ministry of Education (MHRD), India. This means, as x x gets larger and larger, f (x) f (x) gets larger and larger as well. Recall that we call this behavior the end behavior of a function. WebThe graph has 4 turning points, so the lowest degree it can have is degree which is 1 more than the number of turning points 5. From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side. Use the fact above to determine the x x -intercept that corresponds to each zero will cross the x x -axis or just touch it and if the x x -intercept will flatten out or not. Notice in the figure belowthat the behavior of the function at each of the x-intercepts is different. The zero associated with this factor, \(x=2\), has multiplicity 2 because the factor \((x2)\) occurs twice. How to find f(y) = 16y 5 + 5y 4 2y 7 + y 2. (You can learn more about even functions here, and more about odd functions here). How can you tell the degree of a polynomial graph The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis. If a function has a global minimum at \(a\), then \(f(a){\leq}f(x)\) for all \(x\). We can also graphically see that there are two real zeros between [latex]x=1[/latex]and [latex]x=4[/latex]. Identify the x-intercepts of the graph to find the factors of the polynomial. The graph of polynomial functions depends on its degrees. Find the polynomial of least degree containing all the factors found in the previous step. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. If a polynomial contains a factor of the form (x h)p, the behavior near the x-intercept h is determined by the power p. We say that x = h is a zero of multiplicity p. NIOS helped in fulfilling her aspiration, the Board has universal acceptance and she joined Middlesex University, London for BSc Cyber Security and Hence, we already have 3 points that we can plot on our graph. An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Example \(\PageIndex{9}\): Using the Intermediate Value Theorem. Given that f (x) is an even function, show that b = 0. Roots of a polynomial are the solutions to the equation f(x) = 0. For now, we will estimate the locations of turning points using technology to generate a graph. Look at the graph of the polynomial function \(f(x)=x^4x^34x^2+4x\) in Figure \(\PageIndex{12}\). The minimum occurs at approximately the point \((0,6.5)\), WebEx: Determine the Least Possible Degree of a Polynomial The sign of the leading coefficient determines if the graph's far-right behavior. Examine the behavior of the We can see the difference between local and global extrema below. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity. You can get in touch with Jean-Marie at https://testpreptoday.com/. Graphs Each linear expression from Step 1 is a factor of the polynomial function. graduation. To confirm algebraically, we have, \[\begin{align} f(-x) =& (-x)^6-3(-x)^4+2(-x)^2\\ =& x^6-3x^4+2x^2\\ =& f(x). Does SOH CAH TOA ring any bells? The least possible even multiplicity is 2. (2x2 + 3x -1)/(x 1)Variables in thedenominator are notallowed. First, lets find the x-intercepts of the polynomial. A quadratic equation (degree 2) has exactly two roots. Examine the The graphs of \(f\) and \(h\) are graphs of polynomial functions. There are lots of things to consider in this process. WebWe determine the polynomial function, f (x), with the least possible degree using 1) turning points 2) The x-intercepts ("zeros") to find linear factors 3) Multiplicity of each factor 4) Write a formula for the polynomial function. We actually know a little more than that. 2 is a zero so (x 2) is a factor. See Figure \(\PageIndex{13}\). The degree of a polynomial is the highest degree of its terms. The graph of a polynomial function changes direction at its turning points. See Figure \(\PageIndex{8}\) for examples of graphs of polynomial functions with multiplicity \(p=1, p=2\), and \(p=3\). How to find the degree of a polynomial with a graph - Math Index Get math help online by chatting with a tutor or watching a video lesson. A polynomial function of degree \(n\) has at most \(n1\) turning points. How to find the degree of a polynomial The sum of the multiplicities is the degree of the polynomial function.Oct 31, 2021. Get math help online by speaking to a tutor in a live chat. Example \(\PageIndex{2}\): Finding the x-Intercepts of a Polynomial Function by Factoring. We call this a triple zero, or a zero with multiplicity 3. When the leading term is an odd power function, as \(x\) decreases without bound, \(f(x)\) also decreases without bound; as \(x\) increases without bound, \(f(x)\) also increases without bound. Developing a conducive digital environment where students can pursue their 10/12 level, degree and post graduate programs from the comfort of their homes even if they are attending a regular course at college/school or working. This gives us five x-intercepts: \((0,0)\), \((1,0)\), \((1,0)\), \((\sqrt{2},0)\),and \((\sqrt{2},0)\). How to find the degree of a polynomial For zeros with even multiplicities, the graphs touch or are tangent to the x-axis. The degree could be higher, but it must be at least 4. [latex]\begin{array}{l}\hfill \\ f\left(0\right)=-2{\left(0+3\right)}^{2}\left(0 - 5\right)\hfill \\ \text{}f\left(0\right)=-2\cdot 9\cdot \left(-5\right)\hfill \\ \text{}f\left(0\right)=90\hfill \end{array}[/latex]. Additionally, we can see the leading term, if this polynomial were multiplied out, would be \(2x3\), so the end behavior 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. The same is true for very small inputs, say 100 or 1,000. As [latex]x\to -\infty [/latex] the function [latex]f\left(x\right)\to \infty [/latex], so we know the graph starts in the second quadrant and is decreasing toward the, Since [latex]f\left(-x\right)=-2{\left(-x+3\right)}^{2}\left(-x - 5\right)[/latex] is not equal to, At [latex]\left(-3,0\right)[/latex] the graph bounces off of the. Sometimes, a turning point is the highest or lowest point on the entire graph. WebThe degree of a polynomial function affects the shape of its graph. The multiplicity of a zero determines how the graph behaves at the x-intercepts. From the Factor Theorem, we know if -1 is a zero, then (x + 1) is a factor. Get Solution. Graphs of Second Degree Polynomials For the odd degree polynomials, y = x3, y = x5, and y = x7, the graph skims the x-axis in each case as it crosses over the x-axis and also flattens out as the power of the variable increases. Technology is used to determine the intercepts. At x= 3, the factor is squared, indicating a multiplicity of 2. We have already explored the local behavior of quadratics, a special case of polynomials. The sum of the multiplicities is no greater than \(n\). To start, evaluate [latex]f\left(x\right)[/latex]at the integer values [latex]x=1,2,3,\text{ and }4[/latex]. More References and Links to Polynomial Functions Polynomial Functions The graph looks almost linear at this point. Sometimes, a turning point is the highest or lowest point on the entire graph. 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. Write a formula for the polynomial function shown in Figure \(\PageIndex{20}\). For our purposes in this article, well only consider real roots. Use the end behavior and the behavior at the intercepts to sketch the graph. Let us look at P (x) with different degrees. Tap for more steps 8 8. The maximum number of turning points of a polynomial function is always one less than the degree of the function. The graph of a degree 3 polynomial is shown. WebHow To: Given a graph of a polynomial function, write a formula for the function Identify the x -intercepts of the graph to find the factors of the polynomial. The last zero occurs at [latex]x=4[/latex]. These questions, along with many others, can be answered by examining the graph of the polynomial function. exams to Degree and Post graduation level. find degree For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Recall that if \(f\) is a polynomial function, the values of \(x\) for which \(f(x)=0\) are called zeros of \(f\). Since both ends point in the same direction, the degree must be even. 6 has a multiplicity of 1. Getting back to our example problem there are several key points on the graph: the three zeros and the y-intercept. . If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. An example of data being processed may be a unique identifier stored in a cookie. Figure \(\PageIndex{23}\): Diagram of a rectangle with four squares at the corners. A monomial is one term, but for our purposes well consider it to be a polynomial. The factor is repeated, that is, the factor [latex]\left(x - 2\right)[/latex] appears twice. If a function has a local minimum at \(a\), then \(f(a){\leq}f(x)\)for all \(x\) in an open interval around \(x=a\). This gives the volume, \[\begin{align} V(w)&=(202w)(142w)w \\ &=280w68w^2+4w^3 \end{align}\]. If a function has a global maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x. b.Factor any factorable binomials or trinomials. If p(x) = 2(x 3)2(x + 5)3(x 1). This page titled 3.4: Graphs of Polynomial Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax. Figure \(\PageIndex{16}\): The complete graph of the polynomial function \(f(x)=2(x+3)^2(x5)\). How to find The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in Table \(\PageIndex{1}\). The end behavior of a polynomial function depends on the leading term. Fortunately, we can use technology to find the intercepts. Given a polynomial's graph, I can count the bumps. Keep in mind that some values make graphing difficult by hand. My childs preference to complete Grade 12 from Perfect E Learn was almost similar to other children. At \(x=5\),the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept. Lets not bother this time! Using technology, we can create the graph for the polynomial function, shown in Figure \(\PageIndex{16}\), and verify that the resulting graph looks like our sketch in Figure \(\PageIndex{15}\). Step 3: Find the y \end{align}\]. order now. A polynomial function of n th degree is the product of n factors, so it will have at most n roots or zeros, or x -intercepts. WebSince the graph has 3 turning points, the degree of the polynomial must be at least 4. \(\PageIndex{5}\): Given the graph shown in Figure \(\PageIndex{21}\), write a formula for the function shown. Graphs behave differently at various x-intercepts. Your polynomial training likely started in middle school when you learned about linear functions. 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. WebThe degree of a polynomial function helps us to determine the number of x -intercepts and the number of turning points. WebHow to determine the degree of a polynomial graph. The graph crosses the x-axis, so the multiplicity of the zero must be odd. The shortest side is 14 and we are cutting off two squares, so values \(w\) may take on are greater than zero or less than 7. The graph of function \(g\) has a sharp corner. So, the function will start high and end high. For example, a polynomial function of degree 4 may cross the x-axis a maximum of 4 times. Identify the x-intercepts of the graph to find the factors of the polynomial. Sketch the polynomial p(x) = (1/4)(x 2)2(x + 3)(x 5). If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity. Each zero has a multiplicity of one. Step 3: Find the y-intercept of the. Graphical Behavior of Polynomials at x-Intercepts. The graphed polynomial appears to represent the function [latex]f\left(x\right)=\frac{1}{30}\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex]. For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis. The sum of the multiplicities is the degree of the polynomial function. Let \(f\) be a polynomial function. The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below. Do all polynomial functions have a global minimum or maximum? We can apply this theorem to a special case that is useful in graphing polynomial functions. \[\begin{align} f(0)&=a(0+3)(02)^2(05) \\ 2&=a(0+3)(02)^2(05) \\ 2&=60a \\ a&=\dfrac{1}{30} \end{align}\]. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, \(a_nx^n\), is an even power function and \(a_n>0\), as \(x\) increases or decreases without bound, \(f(x)\) increases without bound. In these cases, we say that the turning point is a global maximum or a global minimum. \[\begin{align} h(x)&=x^3+4x^2+x6 \\ &=(x+3)(x+2)(x1) \end{align}\]. 1. n=2k for some integer k. This means that the number of roots of the Write the equation of a polynomial function given its graph. If the graph crosses the x -axis and appears almost linear at the intercept, it is a single zero. How many points will we need to write a unique polynomial? The calculator is also able to calculate the degree of a polynomial that uses letters as coefficients. Sometimes we may not be able to tell the exact power of the factor, just that it is odd or even. To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most \(n1\) turning points. As \(x{\rightarrow}{\infty}\) the function \(f(x){\rightarrow}{\infty}\). Polynomial Function Find the discriminant D of x 2 + 3x + 3; D = 9 - 12 = -3. This leads us to an important idea. The zero of \(x=3\) has multiplicity 2 or 4. How to Find End behavior The Fundamental Theorem of Algebra can help us with that. How do we know if the graph will pass through -3 from above the x-axis or from below the x-axis? 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]. 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. If you need help with your homework, our expert writers are here to assist you. When the leading term is an odd power function, asxdecreases without bound, [latex]f\left(x\right)[/latex] also decreases without bound; as xincreases without bound, [latex]f\left(x\right)[/latex] also increases without bound. In this section we will explore the local behavior of polynomials in general. The graph touches the x-axis, so the multiplicity of the zero must be even. All you can say by looking a graph is possibly to make some statement about a minimum degree of the polynomial. Use the Leading Coefficient Test To Graph To graph a simple polynomial function, we usually make a table of values with some random values of x and the corresponding values of f(x). Polynomial Functions WebThe Fundamental Theorem of Algebra states that, if f(x) is a polynomial of degree n > 0, then f(x) has at least one complex zero. Example \(\PageIndex{4}\): Finding the y- and x-Intercepts of a Polynomial in Factored Form. [latex]{\left(x - 2\right)}^{2}=\left(x - 2\right)\left(x - 2\right)[/latex]. Well, maybe not countless hours. Figure \(\PageIndex{25}\): Graph of \(V(w)=(20-2w)(14-2w)w\). So there must be at least two more zeros. Polynomial factors and graphs | Lesson (article) | Khan Academy The graph has three turning points. Each x-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form. Notice that after a square is cut out from each end, it leaves a \((142w)\) cm by \((202w)\) cm rectangle for the base of the box, and the box will be \(w\) cm tall. Call this point [latex]\left(c,\text{ }f\left(c\right)\right)[/latex]. Step 1: Determine the graph's end behavior. a. We can use this graph to estimate the maximum value for the volume, restricted to values for wthat are reasonable for this problem, values from 0 to 7. Local Behavior of Polynomial Functions For zeros with odd multiplicities, the graphs cross or intersect the x-axis at these x-values. Given the graph below with y-intercept 1.2, write a polynomial of least degree that could represent the graph. A monomial is a variable, a constant, or a product of them. WebThe graph is shown at right using the WINDOW (-5, 5) X (-8, 8). WebRead on for some helpful advice on How to find the degree of a polynomial from a graph easily and effectively. where \(R\) represents the revenue in millions of dollars and \(t\) represents the year, with \(t=6\)corresponding to 2006.
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