Section Exercises

1. If the graphs of two linear functions are parallel, describe the relationship between the slopes and the y-intercepts. 2. If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y-intercepts. 3. If a horizontal line has the equation [latex]f\left(x\right)=a\\[/latex] and a vertical line has the equation [latex]x=a\\[/latex], what is the point of intersection? Explain why what you found is the point of intersection. 4. Explain how to find a line parallel to a linear function that passes through a given point. 5. Explain how to find a line perpendicular to a linear function that passes through a given point. For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular: 6. [latex]\begin{cases}4x - 7y=10\hfill \\ 7x+4y=1\hfill \end{cases}\\[/latex] 7. [latex]\begin{cases}3y+x=12\\ -y=8x+1\end{cases}\\[/latex] 8. [latex]\begin{cases}3y+4x=12\\ -6y=8x+1\end{cases}\\[/latex] 9. [latex]\begin{cases}6x - 9y=10\\ 3x+2y=1\end{cases}\\[/latex] 10. [latex]\begin{cases}y=\frac{2}{3}x+1\\ 3x+2y=1\end{cases}\\[/latex] 11. [latex]\begin{cases}y=\frac{3}{4}x+1\\ -3x+4y=1\end{cases}\\[/latex] For the following exercises, find the x- and y-intercepts of each equation. 12. [latex]f\left(x\right)=-x+2\\[/latex] 13. [latex]g\left(x\right)=2x+4\\[/latex] 14. [latex]h\left(x\right)=3x - 5\\[/latex] 15. [latex]k\left(x\right)=-5x+1\\[/latex] 16. [latex]-2x+5y=20\\[/latex] 17. [latex]7x+2y=56\\[/latex] For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? 18. Line 1: Passes through [latex]\left(0,6\right)\\[/latex] and [latex]\left(3,-24\right)\\[/latex] Line 2: Passes through [latex]\left(-1,19\right)\\[/latex] and [latex]\left(8,-71\right)\\[/latex] 19. Line 1: Passes through [latex]\left(-8,-55\right)\\[/latex] and [latex]\left(10,89\right)\\[/latex] Line 2: Passes through [latex]\left(9,-44\right)\\[/latex] and [latex]\left(4,-14\right)\\[/latex] 20. Line 1: Passes through [latex]\left(2,3\right)\\[/latex] and [latex]\left(4,-1\right)\\[/latex] Line 2: Passes through [latex]\left(6,3\right)\\[/latex] and [latex]\left(8,5\right)\\[/latex] 21. Line 1: Passes through [latex]\left(1,7\right)\\[/latex] and [latex]\left(5,5\right)\\[/latex] Line 2: Passes through [latex]\left(-1,-3\right)\\[/latex] and [latex]\left(1,1\right)\\[/latex] 22. Line 1: Passes through [latex]\left(0,5\right)\\[/latex] and [latex]\left(3,3\right)\\[/latex] Line 2: Passes through [latex]\left(1,-5\right)\\[/latex] and [latex]\left(3,-2\right)\\[/latex] 23. Line 1: Passes through [latex]\left(2,5\right)\\[/latex] and [latex]\left(5,-1\right)\\[/latex] Line 2: Passes through [latex]\left(-3,7\right)\\[/latex] and [latex]\left(3,-5\right)\\[/latex] 24. Write an equation for a line parallel to [latex]f\left(x\right)=-5x - 3\\[/latex] and passing through the point [latex]\left(2,\text{ -}12\right)\\[/latex]. 25. Write an equation for a line parallel to [latex]g\left(x\right)=3x - 1\\[/latex] and passing through the point [latex]\left(4,9\right)\\[/latex]. 26. Write an equation for a line perpendicular to [latex]h\left(t\right)=-2t+4\\[/latex] and passing through the point [latex]\left(\text{-}4,\text{ -}1\right)\\[/latex]. 27. Write an equation for a line perpendicular to [latex]p\left(t\right)=3t+4\\[/latex] and passing through the point [latex]\left(3,1\right)\\[/latex]. 28. Find the point at which the line [latex]f\left(x\right)=-2x - 1\\[/latex] intersects the line [latex]g\left(x\right)=-x\\[/latex]. 29. Find the point at which the line [latex]f\left(x\right)=2x+5\\[/latex] intersects the line [latex]g\left(x\right)=-3x - 5\\[/latex]. 30. Use algebra to find the point at which the line [latex]f\left(x\right)= -\frac{4}{5}x +\frac{274}{25}\\[/latex] intersects the line [latex]h\left(x\right)=\frac{9}{4}x+\frac{73}{10}\\[/latex]. 31. Use algebra to find the point at which the line [latex]f\left(x\right)=\frac{7}{4}x+\frac{457}{60}\\[/latex] intersects the line [latex]g\left(x\right)=\frac{4}{3}x+\frac{31}{5}\\[/latex]. For the following exercises, the given linear equation with its graph. 32. [latex]f\left(x\right)=-x - 1\\[/latex] 33. [latex]f\left(x\right)=-2x - 1\\[/latex] 34. [latex]f\left(x\right)=-\frac{1}{2}x - 1\\[/latex] 35. [latex]f\left(x\right)=2\\[/latex] 36. [latex]f\left(x\right)=2+x\\[/latex] 37. [latex]f\left(x\right)=3x+2\\[/latex] For the following exercises, sketch a line with the given features. 38. An x-intercept of [latex]\left(-\text{4},\text{ 0}\right)\\[/latex] and y-intercept of [latex]\left(0,\text{ -2}\right)\\[/latex] 39. An x-intercept of [latex]\left(-\text{2},\text{ 0}\right)\\[/latex] and y-intercept of [latex]\left(0,\text{ 4}\right)\\[/latex] 40. A y-intercept of [latex]\left(0,\text{ 7}\right)\\[/latex] and slope [latex]-\frac{3}{2}\\[/latex] 41. A y-intercept of [latex]\left(0,\text{ 3}\right)\\[/latex] and slope [latex]\frac{2}{5}\\[/latex] 42. Passing through the points [latex]\left(-\text{6},\text{ -2}\right)\\[/latex] and [latex]\left(\text{6},\text{ -6}\right)\\[/latex] 43. Passing through the points [latex]\left(-\text{3},\text{ -4}\right)\\[/latex] and [latex]\left(\text{3},\text{ 0}\right)\\[/latex] For the following exercises, sketch the graph of each equation. 44. [latex]f\left(x\right)=-2x - 1\\[/latex] 45. [latex]g\left(x\right)=-3x+2\\[/latex] 46. [latex]h\left(x\right)=\frac{1}{3}x+2\\[/latex] 47. [latex]k\left(x\right)=\frac{2}{3}x - 3\\[/latex] 48. [latex]f\left(t\right)=3+2t\\[/latex] 49. [latex]p\left(t\right)=-2+3t\\[/latex] 50. [latex]x=3\\[/latex] 51. [latex]x=-2\\[/latex] 52. [latex]r\left(x\right)=4\\[/latex] 53. [latex]q\left(x\right)=3\\[/latex] 54. [latex]4x=-9y+36\\[/latex] 55. [latex]\frac{x}{3}-\frac{y}{4}=1\\[/latex] 56. [latex]3x - 5y=15\\[/latex] 57. [latex]3x=15\\[/latex] 58. [latex]3y=12\\[/latex] 59. If [latex]g\left(x\right)\\[/latex] is the transformation of [latex]f\left(x\right)=x\\[/latex] after a vertical compression by [latex]\frac{3}{4}\\[/latex], a shift right by 2, and a shift down by 4

a. Write an equation for [latex]g\left(x\right)\\[/latex].

b. What is the slope of this line?

c. Find the y-intercept of this line.

a. Write an equation for [latex]g\left(x\right)\\[/latex].

b. What is the slope of this line?

c. Find the y-intercept of this line.

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