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Graphs of polynomials

Analyze polynomials in order to sketch their graph.

What you should be familiar with before taking this lesson

The end behavior of a function f describes the behavior of its graph at the "ends" of the x-axis. Algebraically, end behavior is determined by the following two questions:
  • As x, right arrow, plus, infinity, what does f, left parenthesis, x, right parenthesis approach?
  • As x, right arrow, minus, infinity, what does f, left parenthesis, x, right parenthesis approach?
If this is new to you, we recommend that you check out our end behavior of polynomials article.
The zeros of a function f correspond to the x-intercepts of its graph. If f has a zero of odd multiplicity, its graph will cross the x-axis at that x value. If f has a zero of even multiplicity, its graph will touch the x-axis at that point.
If this is new to you, we recommend that you check out our zeros of polynomials article.

What you will learn in this lesson

In this lesson, we will use the above features in order to analyze and sketch graphs of polynomials. We will then use the sketch to find the polynomial's positive and negative intervals.

Analyzing polynomial functions

We will now analyze several features of the graph of the polynomial f, left parenthesis, x, right parenthesis, equals, left parenthesis, 3, x, minus, 2, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, squared.

Finding the y-intercept

To find the y-intercept of the graph of f, we can find f, left parenthesis, 0, right parenthesis.
f(x)=(3x2)(x+2)2f(0)=(3(0)2)(0+2)2f(0)=(2)(4)f(0)=8\begin{aligned} f(x)&=(3x-2)(x+2)^2 \\\\ f(\tealD0)&= (3(\tealD 0)-2)(\tealD0+2)^2\\ \\ f(0)&= (-2)(4)\\\\ f(0)&=-8 \end{aligned}
The y-intercept of the graph of y, equals, f, left parenthesis, x, right parenthesis is left parenthesis, 0, comma, minus, 8, right parenthesis.

Finding the x-intercepts

To find the x-intercepts, we can solve the equation f, left parenthesis, x, right parenthesis, equals, 0.
f(x)=(3x2)(x+2)20=(3x2)(x+2)2\begin{aligned} f(x)&=(3x-2)(x+2)^2 \\\\ \tealD 0&= (3x-2)(x+2)^2\\ \\ \end{aligned}
3x2=0orx+2=0Zero product propertyx=23orx=2\begin{aligned}&\swarrow&\searrow\\\\ 3x-2&=0&\text{or}\quad x+2&=0&\small{\gray{\text{Zero product property}}}\\\\ x&=\dfrac{2}{3}&\text{or}\qquad x&=-2\end{aligned}
The x-intercepts of the graph of y, equals, f, left parenthesis, x, right parenthesis are left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis and left parenthesis, minus, 2, comma, 0, right parenthesis.
Our work also shows that start fraction, 2, divided by, 3, end fraction is a zero of multiplicity 1 and minus, 2 is a zero of multiplicity 2. This means that the graph will cross the x-axis at left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis and touch the x-axis at left parenthesis, minus, 2, comma, 0, right parenthesis.

Finding the end behavior

To find the end behavior of a function, we can examine the leading term when the function is written in standard form.
Let's write the equation in standard form.
f(x)=(3x2)(x+2)2f(x)=(3x2)(x2+4x+4)f(x)=3x3+12x2+12x2x28x8f(x)=3x3+10x2+4x8\begin{aligned}f(x)&=(3x-2)(x+2)^2\\ \\ f(x)&=(3x-2)(x^2+4x+4)\\ \\ f(x)&=3x^3+12x^2+12x-2x^2-8x-8\\ \\ f(x)&=\goldD{3x^3}+10x^2+4x-8 \end{aligned}
The leading term of the polynomial is start color #e07d10, 3, x, cubed, end color #e07d10, and so the end behavior of function f will be the same as the end behavior of 3, x, cubed.
Since the degree is odd and the leading coefficient is positive, the end behavior will be: as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity and as x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity.

Sketching a graph

We can use what we've found above to sketch a graph of y, equals, f, left parenthesis, x, right parenthesis.
Let's start with end behavior:
  • As x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity.
  • As x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity.
This means that in the "ends," the graph will look like the graph of y, equals, x, cubed.
Now we can add what we know about the x-intercepts:
  • The graph touches the x-axis at left parenthesis, minus, 2, comma, 0, right parenthesis, since minus, 2 is a zero of even multiplicity.
  • The graph crosses the x-axis at left parenthesis, start fraction, 2, divided by, 3, end fraction, comma, 0, right parenthesis, since start fraction, 2, divided by, 3, end fraction is a zero of odd multiplicity.
Finally, let's finish this process by plotting the y-intercept left parenthesis, 0, comma, minus, 8, right parenthesis and filling in the gaps with a smooth, continuous curve.
While we don't know exactly where the turning points are, we still have a good idea of the overall shape of the function's graph!

Positive and negative intervals

Now that we have a sketch of f's graph, it is easy to determine the intervals for which f is positive, and those for which it is negative.
We see that f is positive when x, is greater than, start fraction, 2, divided by, 3, end fraction and negative when x, is less than, minus, 2 or minus, 2, is less than, x, is less than, start fraction, 2, divided by, 3, end fraction.

Check your understanding

1) You will now work towards a sketch of 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 on your own.
a) What is the y-intercept of the graph of 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?
left parenthesis, 0,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis

b) What is the end behavior of the graph of 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?
Choose 1 answer:
Choose 1 answer:

c) What are the x-intercepts of the graph of 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?
Choose 1 answer:
Choose 1 answer:

d) Which of the following graphs could be the graph of 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?
Choose 1 answer:
Choose 1 answer:

2) Which of the following could be the graph of y, equals, left parenthesis, 2, minus, x, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, squared
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • duskpin seed style avatar for user bavila470
    Can there be any easier explanation of the end behavior please. The way that it was explained in the text, made me get a little confused. Also, for the practice problem, when ever x equals zero, does it mean that we only solve the remaining numbers that are not zeros?
    (5 votes)
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  • blobby green style avatar for user john.cueva
    How can you graph f(x)=x^2 + 2x - 5? step by step?
    (I'm new here)
    (5 votes)
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    • leaf orange style avatar for user A/V
      Given a polynomial in that form, the best way to graph it by hand is to use a table.

      My personal rule is to always use the values most easiest, so let's use -2,-1,0,1,2

      F(-2) = (-2)²-2(-2)-5 = 4+4-5 = 3 ==> (-2,3)
      F(-1) = (-1)²-2(-1)-5 = 1+2-5 = -2 ==> (-1,-2)
      F(0) = (0)² -2(0) -5 = 0-0-5 = -5 ==> (0,-5)
      F(1) = (1)² -2(1) -5 = 1-2-5 ==> (1,-6)
      F(2) = (2)² -2(2)-5 = 4-4-5 = (2,-5)


      ^ just a reminder that the number inside the f(x) (e.g f(-1)) is the number that you plug in the equation.

      As you notice, the value inside the f(x) is the x-coordinate, and the output of that is the y coordinate
      (e.g (-1,-2) at f(-1)).

      hopefully that helps ! and if you have any suspicious, 100% recommend using desmos to check your work.
      (4 votes)
  • piceratops ultimate style avatar for user Cătălin Gherasim Cîrcu
    What throws me off here is the way you gentlemen graphed the Y intercept. Surely there is a reason behind it but for me it is quite unclear why the scale of the y intercept (0,-8) would be the same as (2/3,0). Would appreciate an answer. Have a good day!
    (2 votes)
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    • aqualine tree style avatar for user Judith Gibson
      I see what you mean, but keep in mind that although the scale used on the X-axis is almost always the same as the scale used on the Y-axis, they do not HAVE TO BE the same.
      I can't answer for the designers of the graph, but perhaps they wanted to save space, or perhaps it was done in error --- who knows!
      (7 votes)
  • area 52 yellow style avatar for user Coward
    Question number 2--'which of the following could be a graph for y = (2-x)(x+1)^2' confuses me slightly. Shouldn't the y-intercept be -2?
    y = (2-0)(0+1)(0+1)
    y = (-2)(1)(1)
    y = -2
    What makes the y-intercept positive 2 instead of negative 2?
    (3 votes)
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  • leaf green style avatar for user Tie
    Why were some of the polynomials in factored form?
    (2 votes)
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  • blobby green style avatar for user muhammed
    i cant understand the second question 2) Which of the following could be the graph of y=(2-x)(x+1)^2y=(2−x)(x+1)
    2
    (3 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user ArrowJLC
      Well you could start by looking at the possible zeros. Since the factors are (2-x), (x+1), and (x+1) (because it's squared) then there are two zeros, one at x=2, and the other at x=-1 (because these values make 2-x and x+1 equal to zero).

      Since (x+1) is squared, it has multiplicity 2, which means there's two of them in the factor list. This results in the line of the graph just barely touching zero, rather than crossing it. So you're looking for a graph with zeros at x=-1 and x=2, crossing zero only at x=2.
      Then you determine the end behavior by multiplying all the factors out using algebra, and it has a negative leading coefficient and an odd exponent, which means the end behavior will be x -> (inf) y -> (-inf), and x -> (-inf) y -> (inf).

      Hope this helps!!
      (2 votes)
  • blobby green style avatar for user bdenne14
    How do you match a polynomial function to a graph without being able to use a graphing calculator?
    (2 votes)
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    • leaf blue style avatar for user Stefen
      Seeing and being able to graph a polynomial is an important skill to help develop your intuition of the general behavior of polynomial function. In practice, we rarely graph them since we can tell a lot about what the graph of a polynomial function will look like just by looking at the polynomial itself.
      For example, given ax² + bx + c
      If a is positive, the graph will be like a U and have a minimum value.
      If a is negative, the graph will be flipped and have a maximum value
      If |a| is > 1 the parabola will be very narrow.
      If |a| is < 1 the parabola will be very wide.
      The axis of symmetry (and the location of the vertex) is given by -b/2a.
      When you factor the function, you get the x-intercepts (if any).
      When you set x=0 you get the y intercept of the graph (if any).
      Check out these pages for more information:
      https://www.mathsisfun.com/algebra/quadratic-equation-graphing.html

      Here is a neat tool where you can change the values of the coefficients of a polynomial and see how it affects the resulting graph - a really great way to develop your "mind's eye" and visualize the graph just by looking at the polynomial.
      https://www.mathsisfun.com/algebra/quadratic-equation-graph.html
      (3 votes)
  • blobby green style avatar for user 23gswansonj
    How do you find the end behavior of your graph by just looking at the equation. how do you determine if it is to be flipped?
    (3 votes)
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  • leafers seed style avatar for user allen564
    I get really mixed up with the multiplicity. Is there a video in which someone talks through it?
    (3 votes)
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  • male robot hal style avatar for user Reginato Rezende Moschen
    What is multiplicity of a root and how do I figure out?
    (2 votes)
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    • aqualine ultimate style avatar for user Alissa
      When you have a factor that appears more than once, you can raise that factor to the number power at which it appears. For example if you have (x-4)(x+3)(x-4)(x+1)
      the multiplicities of each factor would be:
      x-4= multiplicity 2
      x+3+ multiplicity 1
      x+1= Multiplicity 1
      (2 votes)