Graphs of Polynomial Functions

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. \(24{x}^{2}-46x+10\)

2. \({t}^{7}-{t}^{3}\)

3. \({y}^{3}+5{y}^{2}-4y-20\)

4. \(12{m}^{2}+20m+3\)

5. \(3{r}^{3}-3\)

6. \(3{m}^{2}-9mn-30{n}^{2}\)

7. \({x}^{3}-{x}^{2}-30x\)

8. Solve \((4x-1)(x+6)=0\)

   Хариулт

   \(\begin{array}{lll}4x-1 & or & x+6=0 \\ 4x=1 & & x=-6 \\ x=\frac{1}{4} & & \end{array}\)

9. \((x-5)(2x+3)=0\)

10. \(x(2x-5)=0\)

11. \({x}^{2}-2x=63\)

    Хариулт

    How is this problem different from practice problems 8 and 9 above? What should be our first step?

    Write the equation in standard form so that one side of the equation is 0.	\({x}^{2}-2x-63=0\)
    Factor the quadratic expression completely.	\((x-9)(x+7)=0\)
    Set each factor containing a variable equal to 0.	\(\begin{array}{lll}x-9=0 & & x+7=0\end{array}\)
    Solve the resulting equations.	\(\begin{array}{lll}x=9 & & x=-7\end{array}\)
    Check each solution in the original equation.	\(\begin{array}{lll}{9}^{2}-2(9)=63 & & {(-7)}^{2}-2(-7)=63\end{array}\)

12. \(4{x}^{2}-25=0\)

13. \(6{x}^{2}=12x\)

14. \(2{x}^{2}-2=-3x\)

15. \((2x+3)(x+1)=15\)

16. \(5f(5f-16)=-15\)

17. Which of the graphs in represents a polynomial function?

    Хариулт

    The graphs of \(f\) and \(h\) are graphs of polynomial functions. They are smooth and continuous.

    The graphs of \(g\) and \(k\) are graphs of functions that are not polynomials. The graph of function \(g\) has a sharp corner. The graph of function \(k\) is not continuous.

18. Find the x-intercepts of \(f(x)={x}^{6}-3{x}^{4}+2{x}^{2}.\)

    Хариулт

    We can attempt to factor this polynomial to find solutions for \(f(x)=0.\)
    

    \[\begin{array}{llll}{x}^{6}-3{x}^{4}+2{x}^{2} & = & 0 & \ \begin{array}{l}\text{Factor out the greatest} \\ \text{common factor}\text{.}\end{array} \\ {x}^{2}({x}^{4}-3{x}^{2}+2) & = & 0 & \ \text{Factor the trinomial}\text{.} \\ {x}^{2}({x}^{2}-1)({x}^{2}-2) & = & 0 & \ \text{Set each factor equal to zero}\text{.}\end{array}\]\[\begin{array}{lllllllllll} & & & & ({x}^{2}-1) & = & 0 & & ({x}^{2}-2) & = & 0 \\ {x}^{2} & = & 0 & \ \text{or} & {x}^{2} & = & 1 & \ \text{or} & {x}^{2} & = & 2 \\ x & = & 0 & & x & = & \pm 1 & & x & = & \pm \sqrt{2}\end{array}\]

    This gives us five x-intercepts: \((0,0),(1,0),(-1,0),(\sqrt{2},0),\) and \((-\sqrt{2},0).\) See . We can see that this is an even function because it is symmetric about the y-axis.

19. Find the x-intercepts of \(f(x)={x}^{3}-5{x}^{2}-x+5.\)

    Хариулт

    Find solutions for \(f(x)=0\) by factoring.
    

    \[\begin{array}{llll}{x}^{3}-5{x}^{2}-x+5 & = & 0 & \ \text{Factor by grouping}\text{.} \\ {x}^{2}(x-5)-(x-5) & = & 0 & \ \text{Factor out the common factor}\text{.} \\ ({x}^{2}-1)(x-5) & = & 0 & \ \text{Factor the difference of squares}\text{.} \\ (x+1)(x-1)(x-5) & = & 0 & \ \text{Set each factor equal to zero}\text{.}\end{array}\]\[\begin{array}{lllllllllll}x+1 & = & 0 & \ \text{or}\ & x-1 & = & 0 & \ \text{or}\ & x-5 & = & 0 \\ x & = & -1 & & x & = & 1 & & x & = & 5\end{array}\]

    There are three x-intercepts: \((-1,0),(1,0),\) and \((5,0).\) See .

20. Find the y- and x-intercepts of \(g(x)={(x-2)}^{2}(2x+3).\)

    Хариулт

    The y-intercept can be found by evaluating \(g(0).\)

    \[\begin{array}{lll}g(0) & = & {(0-2)}^{2}(2(0)+3) \\ & = & 12\end{array}\]

    So the y-intercept is \((0,12).\)

    The x-intercepts can be found by solving \(g(x)=0.\)

    \[{(x-2)}^{2}(2x+3)=0\]\[\begin{array}{lllllll}{(x-2)}^{2} & = & 0 & & (2x+3) & = & 0 \\ x-2 & = & 0 & \ \text{or}\ & x & = & -\frac{3}{2} \\ x & = & 2 & & & & \end{array}\]

    So the x-intercepts are \((2,0)\) and \((-\frac{3}{2},0).\)

21. Find the x-intercepts of \(h(x)={x}^{3}+4{x}^{2}+x-6.\)

    Хариулт

    This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.

    Looking at the graph of this function, as shown in , it appears that there are x-intercepts at \(x=-3,-2,\) and \(1.\)

    We can check whether these are correct by substituting these values for \(x\) and verifying that

    \[h(-3)=h(-2)=h(1)=0\]

    Since \(h(x)={x}^{3}+4{x}^{2}+x-6,\) we have:

    \[\begin{array}{lll}h(-3) & = & {(-3)}^{3}+4{(-3)}^{2}+(-3)-6=-27+36-3-6=0 \\ h(-2) & = & {(-2)}^{3}+4{(-2)}^{2}+(-2)-6=-8+16-2-6=0 \\ h(1) & = & {(1)}^{3}+4{(1)}^{2}+(1)-6=1+4+1-6=0\end{array}\]

    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.

    \[\begin{array}{lll}h(x) & = & {x}^{3}+4{x}^{2}+x-6 \\ & = & (x+3)(x+2)(x-1)\end{array}\]

22. Find the y- and x-intercepts of the function \(f(x)={x}^{4}-19{x}^{2}+30x.\)

    Хариулт

    y-intercept \((0,0);\) x-intercepts \((0,0),(-5,0),(2,0),\) and \((3,0)\)

23. Use the graph of the function of degree 6 in to identify the zeros of the function and their possible multiplicities.

    Хариулт

    The polynomial function is of degree 6. The sum of the multiplicities must be 6.

    Starting from the left, the first zero occurs at \(x=-3.\) The graph touches the x-axis, so the multiplicity of the zero must be even. The zero of \(-3\) most likely has multiplicity \(2.\)

    The next zero occurs at \(x=-1.\) The graph looks almost linear at this point. This is a single zero of multiplicity 1.

    The last zero occurs at \(x=4.\) The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is 6.

24. Use the graph of the function of degree 9 in to identify the zeros of the function and their multiplicities.

    Хариулт

    The graph has a zero of –5 with multiplicity 3, a zero of -1 with multiplicity 2, and a zero of 3 with multiplicity 4.

25. Find the maximum number of turning points of each polynomial function.

    1. ⓐ \(f(x)=-{x}^{3}+4{x}^{5}-3{x}^{2}+1\)
    2. ⓑ \(f(x)=-{(x-1)}^{2}(1+2{x}^{2})\)
    Хариулт

    1. ⓐ

       First, rewrite the polynomial function in descending order: \(f(x)=4{x}^{5}-{x}^{3}-3{x}^{2}+1\)

       Identify the degree of the polynomial function. This polynomial function is of degree 5.

       The maximum number of turning points is \(5-1=4.\)

    2. ⓑ

       First, identify the leading term of the polynomial function if the function were expanded.

       Then, identify the degree of the polynomial function. This polynomial function is of degree 4.

       The maximum number of turning points is \(4-1=3.\)

26. Sketch a graph of \(f(x)=-2{(x+3)}^{2}(x-5).\)

    Хариулт

    This graph has two x-intercepts. At \(x=-3,\) the factor is squared, indicating a multiplicity of 2. The graph will bounce at this x-intercept. At \(x=5,\) the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

    The y-intercept is found by evaluating \(f(0).\)

    \[\begin{array}{lll}f(0) & = & -2{(0+3)}^{2}(0-5) \\ & = & -2⋅9⋅(-5) \\ & = & 90\end{array}\]

    The y-intercept is \((0,90).\)

    Additionally, we can see the leading term, if this polynomial were multiplied out, would be \(-2{x}^{3},\) 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. See .

    To sketch this, we consider that:

    • As \(x\to -\infty\) the function \(f(x)\to \infty ,\) so we know the graph starts in the second quadrant and is decreasing toward the \(x\text{-}\) axis.
    • Since \(f(-x)=-2{(-x+3)}^{2}(-x-5)\) is not equal to \(f(x),\) the graph does not display symmetry.
    • At \((-3,0),\) the graph bounces off of the x-axis, so the function must start increasing.

      At \((0,90),\) the graph crosses the y-axis at the y-intercept. See .

    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 \((5,0).\) See .

    As \(x\to \infty\) the function \(f(x)\to -\infty ,\) so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

    Using technology, we can create the graph for the polynomial function, shown in , and verify that the resulting graph looks like our sketch in .

27. Sketch a graph of \(f(x)=\frac{1}{4}x{(x-1)}^{4}{(x+3)}^{3}.\)

    Хариулт

28. Show that the function \(f(x)={x}^{3}-5{x}^{2}+3x+6\) has at least two real zeros between \(x=1\) and \(x=4.\)

    Хариулт

    As a start, evaluate \(f(x)\) at the integer values \(x=1,2,3,\) and \(4.\) See .

    \(x\)	1	2	3	4
    \(f(x)\)	5	0	–3	2

    We see that one zero occurs at \(x=2.\) 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.

    We have shown that there are at least two real zeros between \(x=1\) and \(x=4.\)

29. Show that the function \(f(x)=7{x}^{5}-9{x}^{4}-{x}^{2}\) has at least one real zero between \(x=1\) and \(x=2.\)

    Хариулт

    Because \(f\) is a polynomial function and since \(f(1)\) is negative and \(f(2)\) is positive, there is at least one real zero between \(x=1\) and \(x=2.\)

30. Write a formula for the polynomial function shown in .

    Хариулт

    This graph has three x-intercepts: \(x=-3,2,\) and \(5.\) The y-intercept is located at \((0,-2).\) At \(x=-3\) and \(x=5,\) the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At \(x=2,\) the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us

    \[f(x)=a(x+3){(x-2)}^{2}(x-5)\]

    To determine the stretch factor, we utilize another point on the graph. We will use the \(y\text{-}\) intercept \((0,-2),\) to solve for \(a.\)

    \[\begin{array}{lll}f(0) & = & a(0+3){(0-2)}^{2}(0-5) \\ -2 & = & a(0+3){(0-2)}^{2}(0-5) \\ -2 & = & -60a \\ a & = & \frac{1}{30}\end{array}\]

    The graphed polynomial appears to represent the function \(f(x)=\frac{1}{30}(x+3){(x-2)}^{2}(x-5).\)

31. Given the graph shown in , write a formula for the function shown.

    Хариулт

    \(f(x)=-\frac{1}{8}{(x-2)}^{3}{(x+1)}^{2}(x-4)\)

32. 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 and then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.

    Хариулт

    We will start this problem by drawing a picture like that in , 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 \((14-2w)\) cm by \((20-2w)\) cm rectangle for the base of the box, and the box will be \(w\) cm tall. This gives the volume

    \[\begin{array}{lll}V(w) & = & (20-2w)(14-2w)w \\ & = & 280w-68{w}^{2}+4{w}^{3}\end{array}\]

    Notice, since the factors are \(w,\) \(20-2w\) and \(14-2w,\) the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider the only the zeros 10 and 7. 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. This means we will restrict the domain of this function to \(0

    From this graph, we turn our focus to only the portion on the reasonable domain, \([0,\ 7].\) We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. 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 .

    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.

33. Use technology to find the maximum and minimum values on the interval \([-1,4]\) of the function \(f(x)=-0.2{(x-2)}^{3}{(x+1)}^{2}(x-4).\)

    Хариулт

    The minimum occurs at approximately the point \((0,-6.5),\) and the maximum occurs at approximately the point \((3.5,7).\)

34. What is the difference between an \(x\text{-}\) intercept and a zero of a polynomial function \(f?\)

    Хариулт

    The \(x\text{-}\) intercept is where the graph of the function crosses the \(x\text{-}\) axis, and the zero of the function is the input value for which \(f(x)=0.\)

35. If a polynomial function of degree \(n\) has \(n\) distinct zeros, what do you know about the graph of the function?

36. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

    Хариулт

    If we evaluate the function at \(a\) and at \(b\) and the sign of the function value changes, then we know a zero exists between \(a\) and \(b.\)

37. Explain how the factored form of the polynomial helps us in graphing it.

38. If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

    Хариулт

    There will be a factor raised to an even power.

39. \(C(t)=2(t-4)(t+1)(t-6)\)

40. \(C(t)=3(t+2)(t-3)(t+5)\)

    Хариулт

    \((-2,0),(3,0),(-5,0)\)

Өөрийнхөөг турш

Parts of this page are adapted from OpenStax College Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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