Chapter 5 Polynomial and Rational Functions
Chapter 5 Review Exercises
Quadratic Functions
For the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts. Finally, graph the function.
1. [latex]f\left(x\right)={x}^{2}-4x-5[/latex]
Show Solution
[latex]f\left(x\right)={\left(x-2\right)}^{2}-9\,\text{vertex} \left(2,–9\right), \text{intercepts} \left(5,0\right); \left(–1,0\right); \left(0,–5\right)[/latex]
2. [latex]f\left(x\right)=-2{x}^{2}-4x[/latex]
For the following exercises, find the equation of the quadratic function using the given information.
3. The vertex is[latex]\left(–2,3\right)[/latex]and a point on the graph is[latex]\,\left(3,6\right).[/latex]
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[latex]f\left(x\right)=\frac{3}{25}{\left(x+2\right)}^{2}+3[/latex]
4. The vertex is[latex]\,\left(–3,6.5\right)\,[/latex]and a point on the graph is[latex]\,\left(2,6\right).[/latex]
For the following exercises, complete the task.
5. A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.
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300 meters by 150 meters, the longer side parallel to the river.
6. An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height,[latex]\,h,\,[/latex]in terms of horizontal distance traveled,[latex]\,x,\,[/latex]given by[latex]\,h\left(x\right)=\frac{-32}{{\left(120\right)}^{2}}{x}^{2}+x.\,[/latex]Find the maximum height the object attains.
Power Functions and Polynomial Functions
For the following exercises, determine if the function is a polynomial function, and if so, give the degree and leading coefficient.
7. [latex]f\left(x\right)=4{x}^{5}-3{x}^{3}+2x-1[/latex]
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Yes, degree = 5, leading coefficient = 4
8. [latex]f\left(x\right)={5}^{x+1}-{x}^{2}[/latex]
9. [latex]f\left(x\right)={x}^{2}\left(3-6x+{x}^{2}\right)[/latex]
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Yes, degree = 4, leading coefficient = 1
For the following exercises, determine end behavior of the polynomial function.
10. [latex]f\left(x\right)=2{x}^{4}+3{x}^{3}-5{x}^{2}+7[/latex]
11. [latex]f\left(x\right)=4{x}^{3}-6{x}^{2}+2[/latex]
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[latex]\text{As}\,x\to -\infty ,\,f\left(x\right)\to -\infty ,\,\text{as}\,x\to \infty ,\,f\left(x\right)\to \infty[/latex]
12. [latex]f\left(x\right)=2{x}^{2}\left(1+3x-{x}^{2}\right)[/latex]
Graphs of Polynomial Functions
For the following exercises, find all zeros of the polynomial function, noting multiplicities.
13. [latex]f\left(x\right)={\left(x+3\right)}^{2}\left(2x-1\right){\left(x+1\right)}^{3}[/latex]
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–3 with multiplicity 2,[latex]-\frac{1}{2}[/latex]with multiplicity 1, –1 with multiplicity 3
14. [latex]f\left(x\right)={x}^{5}+4{x}^{4}+4{x}^{3}[/latex]
15. [latex]f\left(x\right)={x}^{3}-4{x}^{2}+x-4[/latex]
Show Solution
4 with multiplicity 1
For the following exercises, based on the given graph, determine