For the following exercises (1-8), for each pair of points, (a) find the slope of the line passing through the points and (b) indicate whether the line is increasing, decreasing, horizontal, or vertical.
1. [latex](-2,4)[/latex] and [latex](1,1)[/latex]
2. [latex](-1,4)[/latex] and [latex](3,-1)[/latex]
3. [latex](3,5)[/latex] and [latex](-1,2)[/latex]
4. [latex](6,4)[/latex] and [latex](4,-3)[/latex]
5. [latex](2,3)[/latex] and [latex](5,7)[/latex]
6. [latex](1,9)[/latex] and [latex](-8,5)[/latex]
7. [latex](2,4)[/latex] and [latex](1,4)[/latex]
8. [latex](1,4)[/latex] and [latex](1,0)[/latex]
For the following exercises (9-16), write the equation of the line satisfying the given conditions in slope-intercept form.
9. Slope [latex]=-6[/latex], passes through [latex](1,3)[/latex]
10. Slope [latex]=3[/latex], passes through [latex](-3,2)[/latex]
11. Slope [latex]=\dfrac{1}{3}[/latex], passes through [latex](0,4)[/latex]
12. Slope [latex]=\dfrac{2}{5}[/latex], [latex]x[/latex]-intercept [latex]=8[/latex]
13. Passing through [latex](2,1)[/latex] and [latex](-2,-1)[/latex]
14. Passing through [latex](-3,7)[/latex] and [latex](1,2)[/latex]
15. [latex]x[/latex]-intercept [latex]=5[/latex] and [latex]y[/latex]-intercept [latex]=-3[/latex]
16. [latex]x[/latex]-intercept [latex]=-6[/latex] and [latex]y[/latex]-intercept [latex]=9[/latex]
For the following exercises (17-24), for each linear equation, (a) give the slope [latex](m)[/latex], and [latex]y[/latex]-intercept [latex](b)[/latex], if any, and (b) graph the line.
17. [latex]y=2x-3[/latex]
18. [latex]y=-\frac{1}{7}x+1[/latex]
19. [latex]f(x)=-6x[/latex]
20. [latex]f(x)=-5x+4[/latex]
21. [latex]4y+24=0[/latex]
22. [latex]8x-4=0[/latex]
23. [latex]2x+3y=6[/latex]
24. [latex]6x-5y+15=0[/latex]
For the following exercises (25-29), for each polynomial,
- find the degree
- find the zeros, if any
- find the [latex]y[/latex]-intercept(s), if any
- use the leading coefficient to determine the graph’s end behavior
- determine algebraically whether the polynomial is even, odd, or neither.
25. [latex]f(x)=2x^2-3x-5[/latex]
26. [latex]f(x)=-3x^2+6x[/latex]
27. [latex]f(x)=\frac{1}{2}x^2-1[/latex]
28. [latex]f(x)=x^3+3x^2-x-3[/latex]
29. [latex]f(x)=3x-x^3[/latex]
For the following exercises (30-31), use the graph of [latex]f(x)=x^2[/latex] to graph each transformed function [latex]g[/latex].
30. [latex]g(x)=x^2-1[/latex]
31. [latex]g(x)=(x+3)^2+1[/latex]
For the following exercises (32-33), use the graph of [latex]f(x)=\sqrt{x}[/latex] to graph each transformed function [latex]g[/latex].
32. [latex]g(x)=\sqrt{x+2}[/latex]
33. [latex]g(x)=−\sqrt{x}-1[/latex]
For the following exercises (34-35), use the graph of [latex]y=f(x)[/latex] to graph each transformed function [latex]g[/latex]
34. [latex]g(x)=f(x)+1[/latex]
35. [latex]g(x)=f(x-1)+2[/latex]
For the following exercises (36-39), for each of the piecewise-defined functions, (a) evaluate at the given values of the independent variable and (b) sketch the graph.
36. [latex]f(x)=\begin{cases} 4x+3, & x \le 0 \\ -x+1, & x > 0 \end{cases}[/latex]; [latex]f(-3); \, f(0); \, f(2)[/latex]
37. [latex]f(x)=\begin{cases}x^2-3, & x < 0 \\ 4x-3, & x \ge 0 \end{cases}[/latex]; [latex]f(-4); \, f(0); \, f(2)[/latex]
38. [latex]h(x)=\begin{cases} x+1, & x \le 5 \\ 4, & x > 5 \end{cases}[/latex]; [latex]h(0); \, h(\pi ); \, h(5)[/latex]
39. [latex]g(x)=\begin{cases} \left(\dfrac{3}{x-2}\right), & x \ne 2 \\ 4, & x = 2 \end{cases}[/latex]; [latex]g(0); \, g(-4); \, g(2)[/latex]
For the following exercises (40-44), determine whether the statement is true or false. Explain why.
40. [latex]f(x)=\dfrac{(4x+1)}{(7x-2)}[/latex] is a transcendental function.
41. [latex]g(x)=\sqrt[3]{x}[/latex] is an odd root function
42. A logarithmic function is an algebraic function.
43. A function of the form [latex]f(x)=x^b[/latex], where [latex]b[/latex] is a real valued constant, is an exponential function.
44. The domain of an even root function is all real numbers.
45. [T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.
- Find a function [latex]y=V(t)[/latex] that determines the value [latex]V[/latex] of the equipment at the end of [latex]t[/latex] years.
- Find and interpret the meaning of the [latex]x[/latex]– and [latex]y[/latex]-intercepts for this situation.
- What is the value of the equipment at the end of 5 years?
- When will the value of the equipment be $3000?
46. [T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 [latex](t=0)[/latex], total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion.
- Find a linear function [latex]S[/latex] that estimates the total online holiday sales in the year [latex]t[/latex].
- Interpret the slope of the graph of [latex]S[/latex].
- Use part (a) to predict the year when online shopping during Christmas will reach $60 billion.
47. [T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost [latex]C[/latex] as a function of number of cupcakes made.
- Find a linear function that relates cost [latex]C[/latex] to [latex]x[/latex], the number of cupcakes made.
- Find the cost to bake 160 cupcakes.
- If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)
48. [T] A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years.
- Find a linear function that models the price [latex]P[/latex] of the house versus the number of years [latex]t[/latex] since the original purchase.
- Interpret the slope of the graph of [latex]P[/latex].
- Find the price of the house 15 years from when it was originally purchased.
49. [T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year.
- Find a linear function that models the value [latex]V[/latex] of the car after [latex]t[/latex] years.
- Find and interpret [latex]V(4)[/latex].
50. [T] A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.
51. [T] The total cost [latex]C[/latex] (in thousands of dollars) to produce a certain item is modeled by the function [latex]C(x)=10.50x+28,500[/latex], where [latex]x[/latex] is the number of items produced. Determine the cost to produce 175 items.
52. [T] A professor asks her class to report the amount of time [latex]t[/latex] they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.
- Find the linear function [latex]y=N(t)[/latex] that models this situation, where [latex]N[/latex] is the number of pages typed and [latex]t[/latex] is the time in minutes.
- Use part (a) to determine how many pages can be typed in 2 hours.
- Use part (a) to determine how long it takes to type a 20-page assignment.
53. [T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function [latex]P(t)=1.8576t+68.052[/latex], where [latex]t[/latex] is time in years and [latex]t=0[/latex] corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.
54. [T] The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll.
- Find the linear function [latex]y=N(x)[/latex], where [latex]N[/latex] is the number of students that actually enroll and [latex]x[/latex] is the number of all students offered admission to the class of 2019.
- If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.