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.
60. 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{1}{3}[/latex], a shift left by 1, and a shift up by 3a. 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.
For the following exercises, write the equation of the line shown in the graph. 61.
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