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4.10 Sketching polynomials

Lesson

When plotting polynomials the key points to look for are the $x$x-intercepts and $y$y-intercepts, as well as the behaviour of the graph as $x$x goes to $-\infty$ and $\infty$.

As with any graph, the $x$x-intercepts will be the points on the graph where $y=0$y=0 and the $y$y-intercept will be the point on the graph where $x=0$x=0. To find the intercepts it can help to first factorise the polynomial.

Generalising from parabolas, the $x$x-intercepts will be the zeros of the polynomial. That is, if a polynomial can be written as $y=a\left(x-p\right)\left(x-q\right)\left(x-r\right)\dots$y=a(xp)(xq)(xr) then the $x$x-intercepts will be at $x=p,q,r,\dots$x=p,q,r,. Further, the $y$y-intercept will be at $y=apqr\dots$y=apqr. Unlike quadratics, there is not necessarily a turning point at the midpoint of the $x$x-intercepts.

The multiplicity of the zero also matters. The multiplicity is the number of times the factor goes into the polynomial. For example, in $y=\left(x+2\right)^2\left(x-3\right)$y=(x+2)2(x3), $-2$2 has multiplicity $2$2 and $3$3 has multiplicity $1$1. If a zero has multiplicity $2$2 then the graph will touch the $x$x-axis but not cross it. If a zero has multiplicity $3$3 then the graph will have a point of inflection, which means that the graph is horizontal at the point where it crosses the $x$x-axis.

The behaviour of the graph as $x$x goes to $-\infty$ and $\infty$ depends on the leading term of the polynomial. If the degree of the polynomial is odd it will increase on one end and decrease on the other. If it is even it will either increase on both ends or decrease on both ends. If the coefficient is positive the polynomial will increase as $x$x increases and if it is negative it will decrease as $x$x increases.

Summary

The $x$x-intercepts of the graph of a polynomial are at the zeros of the polynomial. The $y$y-intercept will be at the product of the zeros and the constant coefficient.

We can find these either by substituting $y=0$y=0 and $x=0$x=0 or by factorising the polynomial.

The multiplicity of the zero determines the behavious at the $x$x-intercept:

  • Multiplicity $1$1 means the graph crosses the $x$x-axis.
  • Multiplicity $2$2 means the graph touches the $x$x-axis at a turning point.
  • Multiplicity $3$3 means the graph touches the $x$x-axis at a point of inflection, which means that the graph will be horizontal at the $x$x-intercept.

The behaviour of the graph as $x$x goes to $-\infty$ and $\infty$ depends on the leading term of the polynomial:

  • If the degree of the polynomial is odd:
    • If the coefficient of the leading term is positive, the $y$y-values will decrease as $x$x goes to $-\infty$ and increase as $x$x goes to $\infty$
    • If the coefficient of the leading term is negative, the $y$y-values will increase as $x$x goes to $-\infty$ and decrease as $x$x goes to $\infty$
  • If the degree of the polynomial is even:
    • If the coefficient of the leading term is positive, the $y$y-values will increase as $x$x goes to $-\infty$ and increase as $x$x goes to $\infty$
    • If the coefficient of the leading term is negative, the $y$y-values will decrease as $x$x goes to $-\infty$ and decrease as $x$x goes to $\infty$

 

Practice questions

Question 1

The graph of the function $y=f\left(x\right)$y=f(x) is shown below.

Loading Graph...

  1. What are the $x$x-values of the $x$x-intercepts of the graph?

    Enter each $x$x-value on the same line, separated by a comma.

  2. What is the $y$y-value of the $y$y-intercept of the graph?

  3. What is the behaviour of the function as $x\to\infty$x?

    Increasing.

    A

    Decreasing

    B

    Increasing.

    A

    Decreasing

    B
  4. What is the behaviour of the function as $x\to-\infty$x?

    Decreasing

    A

    Increasing

    B

    Decreasing

    A

    Increasing

    B
Question 2

Consider the cubic function $f\left(x\right)=4x^3+8x^2$f(x)=4x3+8x2.

  1. Complete the following table of values.

    $x$x $-6$6 $-2$2 $-1$1 $0$0 $1$1 $2$2
    $f\left(x\right)$f(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Hence sketch the curve $y=f\left(x\right)$y=f(x).

    Loading Graph...

  3. Choose all the correct statements.

    The curve has $x$x intercepts at$-2$2 and $0$0.

    A

    At the point $f\left(2\right)$f(2), the function takes the value of $2$2.

    B

    The curve crosses the $y$y axis at the point $x=0$x=0.

    C

    The value of the function is negative between$x=-2$x=2 and $x=0$x=0.

    D

    At the point $x=2$x=2, the function takes the value of $f\left(2\right)$f(2).

    E

    As the value of $x$x increases from $1$1 to infinity, the curve is increasing.

    F

    The curve has $x$x intercepts at$-2$2 and $0$0.

    A

    At the point $f\left(2\right)$f(2), the function takes the value of $2$2.

    B

    The curve crosses the $y$y axis at the point $x=0$x=0.

    C

    The value of the function is negative between$x=-2$x=2 and $x=0$x=0.

    D

    At the point $x=2$x=2, the function takes the value of $f\left(2\right)$f(2).

    E

    As the value of $x$x increases from $1$1 to infinity, the curve is increasing.

    F
Question 3

A polynomial has been graphed with each of its intercepts shown.

Loading Graph...

  1. Which of the following polynomials could describe the graph shown? (Note, $k$k is a non-zero real number.)

    $y=k\left(x+2\right)^2\left(x-1\right)\left(x-4\right)^2$y=k(x+2)2(x1)(x4)2

    A

    $y=k\left(x+2\right)\left(x-1\right)^3\left(x-4\right)$y=k(x+2)(x1)3(x4)

    B

    $y=k\left(x-2\right)^2\left(x+1\right)\left(x+4\right)^2$y=k(x2)2(x+1)(x+4)2

    C

    $y=k\left(x-2\right)^2\left(x+1\right)^3\left(x+4\right)^2$y=k(x2)2(x+1)3(x+4)2

    D

    $y=k\left(x+2\right)^2\left(x-1\right)\left(x-4\right)^2$y=k(x+2)2(x1)(x4)2

    A

    $y=k\left(x+2\right)\left(x-1\right)^3\left(x-4\right)$y=k(x+2)(x1)3(x4)

    B

    $y=k\left(x-2\right)^2\left(x+1\right)\left(x+4\right)^2$y=k(x2)2(x+1)(x+4)2

    C

    $y=k\left(x-2\right)^2\left(x+1\right)^3\left(x+4\right)^2$y=k(x2)2(x+1)3(x+4)2

    D
  2. Use the $y$y-intercept and the form of the polynomial from part (a) to determine the value of $k$k.

  3. Hence state the equation of the least degree polynomial that could have the graph displayed.

Outcomes

VCMNA361 (10a)

Apply understanding of polynomials to sketch a range of curves and describe the features of these curves from their equation.

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