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Chapter 8: Problem 13

Estimate the maximum number of horizontal intercepts for each of the polynomial functions. Then, using technology, graph the functions to find their approximate values. a. \(y=-2 x^{2}+4 x+3\) b. \(y=\left(t^{2}+1\right)\left(t^{2}-1\right)\) c. \(y=x^{3}+x+1\) d. \(y=x^{5}-3 x^{4}-11 x^{3}+3 x^{2}+10 x\)

Short Answer

Expert verified
a) 2 intercepts: approx. x = -0.58, 2.58. b) 2 intercepts: t = -1, 1. c) 1 intercept: approx. x = -1.325. d) 5 intercepts: approx. x = -3.538, -0.885, 0, 1.881, 5.542.

Step by step solution

01

Understanding Horizontal Intercepts

Horizontal intercepts (or x-intercepts) are the points where the graph of the polynomial crosses the x-axis. To find them, set the polynomial equal to zero and solve for the variable.
02

Determine Degree of Polynomial (Part a)

For the polynomial function \(y = -2x^2 + 4x + 3\), the degree is 2. A polynomial of degree 2 can have at most 2 horizontal intercepts.
03

Graph and Find Horizontal Intercepts (Part a)

Using graphing technology, plot \(y = -2x^2 + 4x + 3\) and identify the points where it crosses the x-axis. Approximate values are found to be \(x = -0.58\) and \(x = 2.58\).
04

Determine Degree of Polynomial (Part b)

For the polynomial function \(y = (t^2 + 1)(t^2 - 1)\), expanding it gives \(y = t^4 - t^2 + t^2 - 1 = t^4 - 1\). The degree is 4. A polynomial of degree 4 can have at most 4 horizontal intercepts.
05

Graph and Find Horizontal Intercepts (Part b)

Using graphing technology, plot \(y = t^4 - 1\). Identified intercepts are at \(t = -1\) and \(t = 1\). There are 2 horizontal intercepts.
06

Determine Degree of Polynomial (Part c)

For the polynomial function \(y = x^3 + x + 1\), the degree is 3. A polynomial of degree 3 can have at most 3 horizontal intercepts.
07

Graph and Find Horizontal Intercepts (Part c)

Using graphing technology, plot \(y = x^3 + x + 1\). Identified intercept is approximately at \(x \approx -1.325\). There is 1 horizontal intercept.
08

Determine Degree of Polynomial (Part d)

For the polynomial function \(y = x^5 - 3x^4 - 11x^3 + 3x^2 + 10x\), the degree is 5. A polynomial of degree 5 can have at most 5 horizontal intercepts.
09

Graph and Find Horizontal Intercepts (Part d)

Using graphing technology, plot \(y = x^5 - 3x^4 - 11x^3 + 3x^2 + 10x\). Identified intercepts are approximately at \(x \approx -3.538, \ -0.885, \ 0, \ 1.881, \ 5.542\). There are 5 horizontal intercepts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Intercepts
Horizontal intercepts, also known as x-intercepts, are key points where a polynomial's graph intersects the x-axis. To find these intercepts, we set the polynomial function equal to zero and solve for the variable. This process tells us the x-values where the function's output is zero. It's essential to remember that the number of horizontal intercepts a polynomial can have is related to its degree, but it might have fewer intercepts in some cases.
Graphing Technology
Using graphing technology, like graphing calculators or software such as Desmos or GeoGebra, can make visualizing polynomials much easier. These tools allow us to input the polynomial function and see its graph. By doing so, we can easily spot where the graph crosses the x-axis, thus identifying the horizontal intercepts. Graphing technology can also provide approximate values for these intercepts, which is extremely helpful when exact analytical solutions are complex or cumbersome to find.
Polynomial Degree
The degree of a polynomial is determined by the highest power of the variable in the polynomial. For example, in the polynomial \(y = x^5 - 3x^4 - 11x^3 + 3x^2 + 10x\), the highest power of x is 5, so it is a degree 5 polynomial. The degree of a polynomial gives us a clue about the maximum number of horizontal intercepts it can have. Specifically, a polynomial of degree n can have up to n horizontal intercepts. However, not every polynomial necessarily meets this maximum; it might have fewer, depending on its specific coefficients and terms.
x-intercepts
Finding x-intercepts involves solving the polynomial equation to see where it equals zero. For example, consider the polynomial \(y = -2x^2 + 4x + 3\). Setting this equation to zero, we solve for x, which provides the x-intercepts at approximately \(x = -0.58\) and \(x = 2.58\). Similarly, for \(y = x^5 - 3x^4 - 11x^3 + 3x^2 + 10x\), the approximate x-intercepts can be found using graphing technology and are \(x \approx -3.538, -0.885, 0, 1.881, 5.542\). Finding these intercepts is a combination of algebraic manipulation and, often, computational tools to get precise values.

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Most popular questions from this chapter

A manufacturer sells children's wooden blocks packed tightly in a cubic tin box with a hinged lid. The blocks cost 3 cents a cubic inch to make. The box and lid material cost 1 cent per square inch. (Assume the sides of the box are so thin that their thickness can be ignored.) It costs 2 cents per linear inch to assemble the box seams. The hinges and clasp on the lid cost \(\$ 2.50,\) and the label costs 50 cents. a. If the edge length of the box is \(s\) inches, develop a formula for estimating the cost \(C(s)\) of making a box that's filled with blocks. b. Graph the function \(C(s)\) for a domain of 0 to \(20 .\) What section of the graph corresponds to what the manufacturer actually produces-boxes between 4 and 16 inches in edge length? c. What is the cost of this product if the cube's edge length is 8 inches? d. Using the graph of \(C(s)\), estimate the edge length of the cube when the total cost is \(\$ 100\)

(Graphing program required.) At low speeds an automobile engine is not at its peak efficiency; efficiency initially rises with speed and then declines at higher speeds. When efficiency is at its maximum, the consumption rate of gas (measured in gallons per hour) is at a minimum. The gas consumption rate of a particular car can be modeled by the following equation, where \(G\) is the gas consumption rate in gallons per hour and \(M\) is speed in miles per hour: \(G=0.0002 M^{2}-0.013 M+1.07\) a. Construct a graph of gas consumption rate versus speed. Estimate the minimum gas consumption rate from your graph and the speed at which it occurs. b. Using the equation for \(G,\) calculate the speed at which the gas consumption rate is at its minimum. What is the minimum gas consumption rate? c. If you travel for 2 hours at peak efficiency, how much gas will you use and how far will you go? d. If you travel at \(60 \mathrm{mph}\), what is your gas consumption rate? How long does it take to go the same distance that you calculated in part (c)? (Recall that travel distance = speed \(\times\) time traveled.) How much gas is required for the trip? e. Compare the answers for parts (c) and (d), which tell you how much gas is used for the same-length trip at two different speeds. Is gas actually saved for the trip by traveling at the speed that gives the minimum gas consumption rate? f. Using the function \(G,\) generate data for gas consumption rate measured in gallons per mile by completing the following table. Plot gallons per mile (on the vertical axis) vs. miles per hour (on the horizontal axis). At what speed is gallons per mile at a minimum? g. Add a fourth column to the data table. This time compute miles/gal \(=\mathrm{mph} /(\mathrm{gal} / \mathrm{hr}) .\) Plot miles per gallon vs. miles per hour. At what speed is miles per gallon at a maximum? This is the inverse of the preceding question; we are normally used to maximizing miles per gallon instead of minimizing gallons per mile. Does your answer make sense in terms of what you found for parts (b) and (f)?

Solve the following equations using the quadratic formula. (Hint: Rewrite each equation so that one side of the equation is zero.) a. \(6 t^{2}-7 t=5\) e. \(6 s^{2}-10=-17 s\) b. \(3 x(3 x-4)=-4\) f. \(2 t^{2}=3 t+9\) c. \((z+1)(3 z-2)=2 z+7\) g. \(5=(4 x+1)(x-3)\) d. \((x+2)(x+4)=1\) h. \((2 x-3)^{2}=7\)

(Graphing program required.) A baseball hit straight up in the air is at a height \(h=4+50 t-16 t^{2}\) feet above ground level at time \(t\) seconds after being hit. (This formula is valid for \(t \geq 0\) until the ball hits the ground.) a. What is the value of \(h\) when \(t=0 ?\) What does this value represent in this context? b. Construct a table of values for \(t=0,1,2,3,4\). Roughly when does the ball hit the ground? How can you tell? c. Graph the function. Does the graph confirm your estimate in part (b)? d. Explain why negative values for \(h\) make no sense in this situation. e. Estimate the maximum height that the baseball reaches. When does it reach that height?

For each part construct a function that satisfies the given conditions. a. Has a constant rate of increase of $$\$ 15,000 /$$ year b. Is a quadratic that opens upward and has a vertex at (1,-4) c. Is a quadratic that opens downward and the vertex is on the \(x\) -axis d. Is a quadratic with a minimum at the point (10,50) and a stretch factor of 3 e. Is a quadratic with a vertical intercept of (0,3) that is also the vertex
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