91Ó°ÊÓ

a. If the degree of a polynomial is odd, then at least one of its zeros must be real. Explain why this is true. b. Sketch a polynomial function that has no real zeros and whose degree is: i. 2 ii. 4 c. Sketch a polynomial function of degree 3 that has exactly: i. One real zero ii. Three real zeros d. Sketch a polynomial function of degree 4 that has exactly two real zeros.

In each part, construct a polynomial function with the indicated characteristics. a. Crosses the \(x\) -axis at least three times b. Crosses the \(x\) -axis at \(-1,3,\) and 10 c. Has a \(y\) -intercept of 4 and degree of 3 d. Has a \(y\) -intercept of -4 and degree of 5

a. Find the equation of the parabola with a vertex of (2,4) that passes through the point (1,7) . b. Construct two different quadratic functions both with a vertex at (2,-3) such that the graph of one function is concave up and the graph of the other function is concave down. c. Find two different equations of a parabola that passes through the points (-2,5) and (4,5) and that opens downward. (Hint: Find the axis of symmetry.)

Let \((h, k)\) be the coordinates of the vertex of a parabola. Then \(h\) is equal to the average of the two real zeros of the function (if they exist). For parts (a) and (b) use this to find \(h\), and then construct an equation in vertex form, \(y=a(x-h)^{2}+k\). a. A parabola with \(x\) -intercepts of 4 and 8 , and a \(y\) -intercept of 32 b. A parabola with \(x\) -intercepts of -3 and \(1,\) and a \(y\) -intercept of -1 c. Can you find the equation of a parabola knowing only its \(x\) -intercepts? Explain.

Let \((h, k)\) be the coordinates of the vertex of a parabola. Then \(h\) equals the average of the two real zeros of the function (if they exist). For each of the following use this to find \(h,\) and then put the equations into the vertex form, \(y=a(x-h)^{2}+k\) a. A parabola with equation \(y=x^{2}+2 x-8\) b. A parabola with equation \(y=-x^{2}-3 x+4\)

a. In economics, revenue \(R\) is defined as the amount of money derived from the sale of a product and is equal to the number \(x\) of units sold times the selling price \(p\) of each unit. What is the equation for revenue? b. If the selling price is given by the equation \(p=-\frac{1}{10} x+20,\) express revenue \(R\) as a function of the number \(x\) of units sold. c. Using technology, plot the function and estimate the number of units that need to be sold to achieve maximum revenue. Then estimate the maximum revenue.

The following tables represent a function \(f\) that converts cups to quarts and a function \(g\) that converts quarts to gallons (all measurements are for fluids). a. Fill in the missing values in the chart. (Hint: One quart contains 4 cups, and one gallon contains 4 quarts.) b. Now evaluate each of the following and identify the units of the results. i. \((g \circ f)(8)\) iii. \(\left(f^{-1} \circ g^{-1}\right)(1)\) ii. \(g^{-1}(2)\) iv. \(\left(f^{-1} \circ g^{-1}\right)(2)\) c. Explain the significance of \(\left(f^{-1} \circ g^{-1}\right)(x)\) in terms of cups, quarts, and gallons.

A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle (see the accompanying figure). If the perimeter of the window is 20 feet (including the semicircle), what dimensions will admit the most light (maximize the area)? (Hint: Express \(L\) in terms of \(r\). Recall that the circumference of a circle \(=2 \pi r,\) and the area of a circle \(=\pi r^{2},\) where \(r\) is the radius of the circle.)

(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)?

A shot-put athlete releases the shot at a speed of 14 meters per second, at an angle of 45 degrees to the horizontal (ground level). The height \(y\) (in meters above the ground) of the shot is given by the function $$ y=2+x-\frac{1}{20} x^{2} $$ where \(x\) is the horizontal distance the shot has traveled (in meters). a. What was the height of the shot at the moment of release? b. How high is the shot after it has traveled 4 meters horizontally from the release point? 16 meters? c. Find the highest point reached by the shot in its flight. d. Draw a sketch of the height of the shot and indicate how far the shot is from the athlete when it lands.