negative leading coefficient graph

The standard form is useful for determining how the graph is transformed from the graph of $y=x^2$. If $a<0$, the parabola opens downward, and the vertex is a maximum. ( Direct link to Reginato Rezende Moschen's post What is multiplicity of a, Posted 5 years ago. We now have a quadratic function for revenue as a function of the subscription charge. We now have a quadratic function for revenue as a function of the subscription charge. If the leading coefficient is negative, bigger inputs only make the leading term more and more negative. If this is new to you, we recommend that you check out our. As with any quadratic function, the domain is all real numbers. step by step? The maximum value of the function is an area of 800 square feet, which occurs when $L=20$ feet. This parabola does not cross the x-axis, so it has no zeros. The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In this form, $a=1$, $b=4$, and $c=3$. From this we can find a linear equation relating the two quantities. Direct link to allen564's post I get really mixed up wit, Posted 3 years ago. Find the vertex of the quadratic function $f(x)=2x^26x+7$. axis of symmetry the function that describes a parabola, written in the form $f(x)=a(xh)^2+k$, where $(h, k)$ is the vertex. A coordinate grid has been superimposed over the quadratic path of a basketball in Figure $\PageIndex{8}$. As x gets closer to infinity and as x gets closer to negative infinity. Write an equation for the quadratic function $g$ in Figure $\PageIndex{7}$ as a transformation of $f(x)=x^2$, and then expand the formula, and simplify terms to write the equation in general form. Since $a$ is the coefficient of the squared term, $a=2$, $b=80$, and $c=0$. Since the leading coefficient is negative, the graph falls to the right. $\PageIndex{5}$: A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. \[\begin{align} Q&=2500p+b &\text{Substitute in the point $Q=84,000$ and $p=30$} \\ 84,000&=2500(30)+b &\text{Solve for $b$} \\ b&=159,000 \end{align}\]. Both ends of the graph will approach negative infinity. We know that currently $p=30$ and $Q=84,000$. If the leading coefficient is negative and the exponent of the leading term is odd, the graph rises to the left and falls to the right. Remember: odd - the ends are not together and even - the ends are together. If $k>0$, the graph shifts upward, whereas if $k<0$, the graph shifts downward. 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. Well you could try to factor 100. We can see the maximum revenue on a graph of the quadratic function. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. Determine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown in Figure $\PageIndex{3}$. Whether you need help with a product or just have a question, our customer support team is always available to lend a helping hand. See Figure $\PageIndex{15}$. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. \[\begin{align} h&=\dfrac{b}{2a} \\ &=\dfrac{9}{2(-5)} \\ &=\dfrac{9}{10} \end{align}\], \[\begin{align} f(\dfrac{9}{10})&=5(\dfrac{9}{10})^2+9(\dfrac{9}{10})-1 \\&= \dfrac{61}{20}\end{align}\]. = Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. The function is an even degree polynomial with a negative leading coefficient Therefore, y + as x -+ Since all of the terms of the function are of an even degree, the function is an even function. Direct link to Seth's post For polynomials without a, Posted 6 years ago. Sketch the graph of the function y = 214 + 81-2 What do we know about this function? Since the degree is odd and the leading coefficient is positive, the end behavior will be: as, We can use what we've found above to sketch a graph of, This means that in the "ends," the graph will look like the graph of. This formula is an example of a polynomial function. ) In terms of end behavior, it also will change when you divide by x, because the degree of the polynomial is going from even to odd or odd to even with every division, but the leading coefficient stays the same. The ball reaches the maximum height at the vertex of the parabola. A polynomial labeled y equals f of x is graphed on an x y coordinate plane. Direct link to loumast17's post End behavior is looking a. Substituting the coordinates of a point on the curve, such as $(0,1)$, we can solve for the stretch factor. Standard or vertex form is useful to easily identify the vertex of a parabola. This problem also could be solved by graphing the quadratic function. 1 This gives us the linear equation $Q=2,500p+159,000$ relating cost and subscribers. It curves back up and passes through the x-axis at (two over three, zero). Direct link to InnocentRealist's post It just means you don't h, Posted 5 years ago. The vertex is at $(2, 4)$. See Table $\PageIndex{1}$. \[t=\dfrac{80-\sqrt{8960}}{32} 5.458 \text{ or }t=\dfrac{80+\sqrt{8960}}{32} 0.458 \]. Does the shooter make the basket? Example $\PageIndex{2}$: Writing the Equation of a Quadratic Function from the Graph. If $a<0$, the parabola opens downward, and the vertex is a maximum. It would be best to put the terms of the polynomial in order from greatest exponent to least exponent before you evaluate the behavior. For the x-intercepts, we find all solutions of $f(x)=0$. When you have a factor that appears more than once, you can raise that factor to the number power at which it appears. We see that f f is positive when x>\dfrac {2} {3} x > 32 and negative when x<-2 x < 2 or -2<x<\dfrac23 2 < x < 32. Either form can be written from a graph. You can see these trends when you look at how the curve y = ax 2 moves as "a" changes: As you can see, as the leading coefficient goes from very . We're here for you 24/7. A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. Find the vertex of the quadratic function $f(x)=2x^26x+7$. Seeing and being able to graph a polynomial is an important skill to help develop your intuition of the general behavior of polynomial function. the function that describes a parabola, written in the form $f(x)=ax^2+bx+c$, where $a,b,$ and $c$ are real numbers and a0. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. Then, to tell desmos to compute a quadratic model, type in y1 ~ a x12 + b x1 + c. You will get a result that looks like this: You can go to this problem in desmos by clicking https://www.desmos.com/calculator/u8ytorpnhk. odd degree with negative leading coefficient: the graph goes to +infinity for large negative values. It is a symmetric, U-shaped curve. How do you find the end behavior of your graph by just looking at the equation. With respect to graphing, the leading coefficient "a" indicates how "fat" or how "skinny" the parabola will be. Can a coefficient be negative? To find the maximum height, find the y-coordinate of the vertex of the parabola. Recall that we find the y-intercept of a quadratic by evaluating the function at an input of zero, and we find the x-intercepts at locations where the output is zero. In this lesson, you will learn what the "end behavior" of a polynomial is and how to analyze it from a graph or from a polynomial equation. We now return to our revenue equation. But if $|a|<1$, the point associated with a particular x-value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. Definition: Domain and Range of a Quadratic Function. For polynomials without a constant term, dividing by x will make a new polynomial, with a degree of n-1, that is undefined at 0. Determine a quadratic functions minimum or maximum value. So the x-intercepts are at $(\frac{1}{3},0)$ and $(2,0)$. The leading coefficient of a polynomial helps determine how steep a line is. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side. Curved antennas, such as the ones shown in Figure $\PageIndex{1}$, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. This allows us to represent the width, $W$, in terms of $L$. 1 Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions. The slope will be, \[\begin{align} m&=\dfrac{79,00084,000}{3230} \\ &=\dfrac{5,000}{2} \\ &=2,500 \end{align}\]. The graph looks almost linear at this point. f, left parenthesis, x, right parenthesis, f, left parenthesis, x, right parenthesis, equals, left parenthesis, 3, x, minus, 2, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, squared, f, left parenthesis, 0, right parenthesis, y, equals, f, left parenthesis, x, right parenthesis, left parenthesis, 0, comma, minus, 8, right parenthesis, f, left parenthesis, x, right parenthesis, equals, 0, left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis, left parenthesis, minus, 2, comma, 0, right parenthesis, start fraction, 2, divided by, 3, end fraction, start color #e07d10, 3, x, cubed, end color #e07d10, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, x, is greater than, start fraction, 2, divided by, 3, end fraction, minus, 2, is less than, x, is less than, start fraction, 2, divided by, 3, end fraction, g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, plus, 5, right parenthesis, g, left parenthesis, x, right parenthesis, right arrow, plus, infinity, g, left parenthesis, x, right parenthesis, right arrow, minus, infinity, left parenthesis, 1, comma, 0, right parenthesis, left parenthesis, 5, comma, 0, right parenthesis, left parenthesis, minus, 1, comma, 0, right parenthesis, left parenthesis, 2, comma, 0, right parenthesis, left parenthesis, minus, 5, comma, 0, right parenthesis, y, equals, left parenthesis, 2, minus, x, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, squared. In finding the vertex, we must be . However, there are many quadratics that cannot be factored. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, $(2,1)$. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down. We find the y-intercept by evaluating $f(0)$. I need so much help with this. degree of the polynomial Find the domain and range of $f(x)=5x^2+9x1$. To find the price that will maximize revenue for the newspaper, we can find the vertex. n You have an exponential function. A horizontal arrow points to the left labeled x gets more negative. It crosses the $y$-axis at $(0,7)$ so this is the y-intercept. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet. If $h>0$, the graph shifts toward the right and if $h<0$, the graph shifts to the left. Given the equation $g(x)=13+x^26x$, write the equation in general form and then in standard form. We can see that if the negative weren't there, this would be a quadratic with a leading coefficient of 1 1 and we might attempt to factor by the sum-product. 2. To determine the end behavior of a polynomial f f from its equation, we can think about the function values for large positive and large negative values of x x. Let's write the equation in standard form. Given a quadratic function in general form, find the vertex of the parabola. On desmos, type the data into a table with the x-values in the first column and the y-values in the second column. When does the rock reach the maximum height? Quadratic functions are often written in general form. The axis of symmetry is $x=\frac{4}{2(1)}=2$. $g(x)=x^26x+13$ in general form; $g(x)=(x3)^2+4$ in standard form. The top part of both sides of the parabola are solid. Direct link to Kim Seidel's post FYI you do not have a , Posted 5 years ago. We can see that the vertex is at $(3,1)$. \[\begin{align} 1&=a(0+2)^23 \\ 2&=4a \\ a&=\dfrac{1}{2} \end{align}\]. Because $a<0$, the parabola opens downward. Direct link to ArrowJLC's post Well you could start by l, Posted 3 years ago. Direct link to A/V's post Given a polynomial in tha, Posted 6 years ago. The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. Substitute the values of the horizontal and vertical shift for $h$ and $k$. Next, select $\mathrm{TBLSET}$, then use $\mathrm{TblStart=6}$ and $\mathrm{Tbl = 2}$, and select $\mathrm{TABLE}$. Given an application involving revenue, use a quadratic equation to find the maximum. This also makes sense because we can see from the graph that the vertical line $x=2$ divides the graph in half. \[\begin{align} h &= \dfrac{80}{2(16)} \\ &=\dfrac{80}{32} \\ &=\dfrac{5}{2} \\ & =2.5 \end{align}\]. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. In either case, the vertex is a turning point on the graph. If the leading coefficient is positive and the exponent of the leading term is even, the graph rises to the left Example $\PageIndex{3}$: Finding the Vertex of a Quadratic Function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power. Would appreciate an answer. Since $xh=x+2$ in this example, $h=2$. { "7.01:_Introduction_to_Modeling" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.02:_Modeling_with_Linear_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.03:_Fitting_Linear_Models_to_Data" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.04:_Modeling_with_Exponential_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.05:_Fitting_Exponential_Models_to_Data" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.06:_Putting_It_All_Together" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.07:_Modeling_with_Quadratic_Functions" : "property get [Map 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\newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$, Example $\PageIndex{1}$: Identifying the Characteristics of a Parabola, Definitions: Forms of Quadratic Functions, HOWTO: Write a quadratic function in a general form, Example $\PageIndex{2}$: Writing the Equation of a Quadratic Function from the Graph, Example $\PageIndex{3}$: Finding the Vertex of a Quadratic Function, Example $\PageIndex{5}$: Finding the Maximum Value of a Quadratic Function, Example $\PageIndex{6}$: Finding Maximum Revenue, Example $\PageIndex{10}$: Applying the Vertex and x-Intercepts of a Parabola, Example $\PageIndex{11}$: Using Technology to Find the Best Fit Quadratic Model, Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions, Determining the Maximum and Minimum Values of Quadratic Functions, https://www.desmos.com/calculator/u8ytorpnhk, source@https://openstax.org/details/books/precalculus, status page at https://status.libretexts.org, Understand how the graph of a parabola is related to its quadratic function, Solve problems involving a quadratic functions minimum or maximum value. The graph of a quadratic function is a parabola. What throws me off here is the way you gentlemen graphed the Y intercept. Solve the quadratic equation $f(x)=0$ to find the x-intercepts. Identify the vertical shift of the parabola; this value is $k$. Setting the constant terms equal: \[\begin{align*} ah^2+k&=c \\ k&=cah^2 \\ &=ca\Big(\dfrac{b}{2a}\Big)^2 \\ &=c\dfrac{b^2}{4a} \end{align*}\]. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Example $\PageIndex{4}$: Finding the Domain and Range of a Quadratic Function. f That is, if the unit price goes up, the demand for the item will usually decrease. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, $p=32$ and $Q=79,000$. See Figure $\PageIndex{16}$. Rewriting into standard form, the stretch factor will be the same as the $a$ in the original quadratic. We can see that the vertex is at $(3,1)$. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. Direct link to 335697's post Off topic but if I ask a , Posted a year ago. Direct link to MonstersRule's post This video gives a good e, Posted 2 years ago. The short answer is yes! What are the end behaviors of sine/cosine functions? A cubic function is graphed on an x y coordinate plane. Math Homework Helper. The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. End behavior is looking at the two extremes of x. Given a quadratic function, find the domain and range. Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function So, there is no predictable time frame to get a response. \[\begin{align} \text{maximum revenue}&=2,500(31.8)^2+159,000(31.8) \\ &=2,528,100 \end{align}\]. So the graph of a cube function may have a maximum of 3 roots. The graph of a quadratic function is a parabola. (credit: Matthew Colvin de Valle, Flickr). Instructors are independent contractors who tailor their services to each client, using their own style, Can there be any easier explanation of the end behavior please. The graph curves down from left to right passing through the origin before curving down again. and the What if you have a funtion like f(x)=-3^x? The range is $f(x){\geq}\frac{8}{11}$, or $\left[\frac{8}{11},\infty\right)$. The infinity symbol throws me off and I don't think I was ever taught the formula with an infinity symbol. If we divided x+2 by x, now we have x+(2/x), which has an asymptote at 0. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. n A part of the polynomial is graphed curving up to touch (negative two, zero) before curving back down. methods and materials. A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. As x\rightarrow -\infty x , what does f (x) f (x) approach? The degree of a polynomial expression is the the highest power (expon. The exponent says that this is a degree- 4 polynomial; 4 is even, so the graph will behave roughly like a quadratic; namely, its graph will either be up on both ends or else be down on both ends. If the parabola opens up, $a>0$. a. Since the sign on the leading coefficient is negative, the graph will be down on both ends. The solutions to the equation are $x=\frac{1+i\sqrt{7}}{2}$ and $x=\frac{1-i\sqrt{7}}{2}$ or $x=\frac{1}{2}+\frac{i\sqrt{7}}{2}$ and $x=\frac{-1}{2}\frac{i\sqrt{7}}{2}$. The unit price of an item affects its supply and demand. We can also confirm that the graph crosses the x-axis at $\Big(\frac{1}{3},0\Big)$ and $(2,0)$. The graph of a . The other end curves up from left to right from the first quadrant. The graph will rise to the right. 1 If you're seeing this message, it means we're having trouble loading external resources on our website. ) \[\begin{align} g(x)&=\dfrac{1}{2}(x+2)^23 \\ &=\dfrac{1}{2}(x+2)(x+2)3 \\ &=\dfrac{1}{2}(x^2+4x+4)3 \\ &=\dfrac{1}{2}x^2+2x+23 \\ &=\dfrac{1}{2}x^2+2x1 \end{align}\]. Example $\PageIndex{10}$: Applying the Vertex and x-Intercepts of a Parabola. This page titled 5.2: Quadratic Functions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. We can use the general form of a parabola to find the equation for the axis of symmetry. The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. Direct link to Stefen's post Seeing and being able to , Posted 6 years ago. The path passes through the origin and has vertex at $(4, 7)$, so $h(x)=\frac{7}{16}(x+4)^2+7$. Notice in Figure $\PageIndex{13}$ that the number of x-intercepts can vary depending upon the location of the graph. Find $h$, the x-coordinate of the vertex, by substituting $a$ and $b$ into $h=\frac{b}{2a}$. The parts of the polynomial are connected by dashed portions of the graph, passing through the y-intercept. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. Direct link to Sirius's post What are the end behavior, Posted 4 months ago. It curves down through the positive x-axis. These features are illustrated in Figure $\PageIndex{2}$. Example $\PageIndex{5}$: Finding the Maximum Value of a Quadratic Function. sinusoidal functions will repeat till infinity unless you restrict them to a domain. A parabola is a U-shaped curve that can open either up or down. Substitute $x=h$ into the general form of the quadratic function to find $k$. . Given a quadratic function, find the x-intercepts by rewriting in standard form. When does the ball reach the maximum height? a The highest power is called the degree of the polynomial, and the . polynomial function The vertex is at $(2, 4)$. Well, let's start with a positive leading coefficient and an even degree. x In this form, $a=3$, $h=2$, and $k=4$. If the parabola has a maximum, the range is given by $f(x){\leq}k$, or $\left(\infty,k\right]$. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of $x$ at which $y=0$. where $(h, k)$ is the vertex. Comment Button navigates to signup page (1 vote) Upvote. One reason we may want to identify the vertex of the parabola is that this point will inform us what the maximum or minimum value of the function is, $(k)$,and where it occurs, $(h)$. The general form of a quadratic function presents the function in the form. The way that it was explained in the text, made me get a little confused. It is labeled As x goes to positive infinity, f of x goes to positive infinity. Given an application involving revenue, use a quadratic equation to find the maximum. It is also helpful to introduce a temporary variable, $W$, to represent the width of the garden and the length of the fence section parallel to the backyard fence. Given the equation $g(x)=13+x^26x$, write the equation in general form and then in standard form. Example. But what about polynomials that are not monomials? + So the axis of symmetry is $x=3$. Next, select $\mathrm{TBLSET}$, then use $\mathrm{TblStart=6}$ and $\mathrm{Tbl = 2}$, and select $\mathrm{TABLE}$. Figure $\PageIndex{8}$: Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes. See Figure $\PageIndex{16}$. Direct link to Katelyn Clark's post The infinity symbol throw, Posted 5 years ago. One important feature of the graph is that it has an extreme point, called the vertex. Posted 7 years ago. f, left parenthesis, x, right parenthesis, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, y, equals, g, left parenthesis, x, right parenthesis, g, left parenthesis, x, right parenthesis, right arrow, plus, infinity, g, left parenthesis, x, right parenthesis, right arrow, minus, infinity, y, equals, a, x, start superscript, n, end superscript, f, left parenthesis, x, right parenthesis, equals, x, squared, g, left parenthesis, x, right parenthesis, equals, minus, 3, x, squared, g, left parenthesis, x, right parenthesis, h, left parenthesis, x, right parenthesis, equals, x, cubed, h, left parenthesis, x, right parenthesis, j, left parenthesis, x, right parenthesis, equals, minus, 2, x, cubed, j, left parenthesis, x, right parenthesis, left parenthesis, start color #11accd, n, end color #11accd, right parenthesis, left parenthesis, start color #1fab54, a, end color #1fab54, right parenthesis, f, left parenthesis, x, right parenthesis, equals, start color #1fab54, a, end color #1fab54, x, start superscript, start color #11accd, n, end color #11accd, end superscript, start color #11accd, n, end color #11accd, start color #1fab54, a, end color #1fab54, is greater than, 0, start color #1fab54, a, end color #1fab54, is less than, 0, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, point, g, left parenthesis, x, right parenthesis, equals, 8, x, cubed, g, left parenthesis, x, right parenthesis, equals, minus, 3, x, squared, plus, 7, x, start color #1fab54, minus, 3, end color #1fab54, x, start superscript, start color #11accd, 2, end color #11accd, end superscript, left parenthesis, start color #11accd, 2, end color #11accd, right parenthesis, left parenthesis, start color #1fab54, minus, 3, end color #1fab54, right parenthesis, f, left parenthesis, x, right parenthesis, equals, 8, x, start superscript, 5, end superscript, minus, 7, x, squared, plus, 10, x, minus, 1, g, left parenthesis, x, right parenthesis, equals, minus, 6, x, start superscript, 4, end superscript, plus, 8, x, cubed, plus, 4, x, squared, start color #ca337c, minus, 3, comma, 000, comma, 000, end color #ca337c, start color #ca337c, minus, 2, comma, 993, comma, 000, end color #ca337c, start color #ca337c, minus, 300, comma, 000, comma, 000, end color #ca337c, start color #ca337c, minus, 290, comma, 010, comma, 000, end color #ca337c, h, left parenthesis, x, right parenthesis, equals, minus, 8, x, cubed, plus, 7, x, minus, 1, g, left parenthesis, x, right parenthesis, equals, left parenthesis, 2, minus, 3, x, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, squared, What determines the rise and fall of a polynomial. 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