Lines and Functions Practice

A point is written (x, y): x says how far across, y how far up or down. The two axes split the plane into four quadrants, numbered I to IV, counterclockwise from the upper right.

1.Quadrants

In quadrant I both numbers are positive. In II, x is negative. In III, both are negative. In IV, y is negative. A point with a 0 sits on an axis.

Name the quadrant, or the axis.

1. 1.Where is (3, 3)?
2. 2.Where is (0, 3)?
3. 3.Where is (1, 6)?
4. 4.Where is (8, 0)?
5. 5.Where is (−5, 1)?
6. 6.Where is (6, −2)?
7. 7.Where is (−7, 6)?
8. 8.Where is (8, −8)?
9. 9.Where is (−6, −7)?
10. 10.Where is (−3, −8)?

2.Distance between two points

Find how far across and how far up. When one of them is 0, the other is the distance. Otherwise, use a² + b² = c².

Find each distance.

1. 11.The distance from (−4, 4) to (0, 1)
2. 12.The distance from (1, 4) to (13, 9)
3. 13.The distance from (−3, 1) to (−3, −2)
4. 14.The distance from (−5, 1) to (−5, −1)
5. 15.The distance from (−5, −1) to (−1, 2)
6. 16.The distance from (3, −6) to (13, −6)
7. 17.The distance from (−1, 6) to (−9, −9)
8. 18.The distance from (6, −6) to (13, −6)
9. 19.The distance from (−6, −6) to (−3, −10)
10. 20.The distance from (−1, −6) to (−1, −14)

3.Midpoints

The midpoint is the average of the two x’s and the average of the two y’s.

Find each midpoint.

1. 21.The midpoint of (1, 5) and (3, 8)
2. 22.The midpoint of (5, 3) and (−2, 4)
3. 23.The midpoint of (7, 4) and (2, −7)
4. 24.The midpoint of (−3, 2) and (7, 3)
5. 25.The midpoint of (−7, 7) and (−4, 4)
6. 26.The midpoint of (−5, −9) and (0, 9)
7. 27.The midpoint of (−5, 1) and (1, −9)
8. 28.The midpoint of (−8, −9) and (−2, −9)

Slope is how steep a line is: the rise divided by the run, or the change in y divided by the change in x.

4.Slope from two points

Subtract the y’s, subtract the x’s in the same order, and divide. A flat line has slope 0. An upright line has no slope at all; its slope is called undefined.

Find each slope.

1. 29.The slope through (3, 6) and (5, 9)
2. 30.The slope through (5, 7) and (10, 10)
3. 31.The slope through (8, −6) and (11, 0)
4. 32.The slope through (−7, 1) and (−1, 2)
5. 33.The slope through (4, 0) and (−3, −7)
6. 34.The slope through (−1, 8) and (−5, 3)
7. 35.The slope through (8, −1) and (12, 1)
8. 36.The slope through (−5, −6) and (2, −2)
9. 37.The slope through (−7, 1) and (−1, 10)
10. 38.The slope through (−8, −5) and (−5, 1)
11. 39.The slope through (−7, 6) and (−11, 13)
12. 40.The slope through (−3, −2) and (−10, 5)

5.Reading y = mx + b

In y = mx + b, m is the slope and b is the y-intercept, where the line crosses the y-axis.

Give the slope and the y-intercept.

1. 41.y = x + 5
2. 42.y = −¹⁄₃x + 3
3. 43.y = 2x − 3
4. 44.y = −5x + 5
5. 45.y = ²⁄₃x − 5
6. 46.y = −⁵⁄₄x + 6
7. 47.y = x − 8
8. 48.y = −6x − 6

6.Slope from standard form

Solve for y first. Then read m and b.

Give the slope and the y-intercept.

1. 49.−x + 4y = 0
2. 50.x − 5y = 5
3. 51.3x − 2y = 6
4. 52.−3x − y = 5
5. 53.−x + 2y = 10
6. 54.2x + 2y = −12
7. 55.−3x − 4y = −8
8. 56.−x + 5y = −15

Every answer in this part is written in the form y = mx + b, unless it asks for something else.

7.From a slope and a point

Put the slope and the point into y = mx + b, and solve for b.

Write the equation.

1. 57.Slope 3, through (4, 2)
2. 58.Slope 1, through (−3, 0)
3. 59.Slope ¹⁄₂, through (0, 8)
4. 60.Slope −5, through (0, 0)
5. 61.Slope ⁴⁄₃, through (6, −7)
6. 62.Slope −2, through (0, 8)
7. 63.Slope −3, through (2, 8)
8. 64.Slope −2, through (−2, 4)

8.From two points

Find the slope first. Then use either point to find b.

Write the equation.

1. 65.Through (1, 7) and (2, 8)
2. 66.Through (−4, 2) and (2, 8)
3. 67.Through (−1, 0) and (1, −4)
4. 68.Through (1, 6) and (3, 14)
5. 69.Through (−2, −5) and (1, 7)
6. 70.Through (−4, 9) and (3, −5)
7. 71.Through (4, −8) and (5, −10)
8. 72.Through (−4, −15) and (5, 12)

9.Intercepts

For the x-intercept, set y = 0 and solve. For the y-intercept, set x = 0 and solve.

Find both intercepts.

1. 73.2x + y = −2
2. 74.x + 7y = 7
3. 75.7x + y = −7
4. 76.4x + y = −8
5. 77.x − 6y = −6
6. 78.2x − 3y = 12
7. 79.7x + 3y = −21
8. 80.8x + 7y = 56

10.Parallel, perpendicular or neither

Parallel lines have the same slope. Perpendicular lines meet at a square corner, and their slopes multiply to −1: 2 and −¹⁄₂, for example.

Decide, and give the reason.

1. 81.y = x + 4 and y = −x + 3
2. 82.y = −⁵⁄₃x + 1 and y = ⁵⁄₃x + 3
3. 83.y = −3x + 7 and y = ¹⁄₃x + 4
4. 84.y = −¹⁄₃x − 3 and y = ²⁄₃x + 6
5. 85.y = 3x − 3 and y = 3x − 7
6. 86.y = −¹⁄₂x − 1 and y = ¹⁄₂x − 4
7. 87.y = −⁵⁄₃x − 2 and y = ³⁄₅x − 3
8. 88.y = −x − 7 and y = −x + 2

11.Parallel and perpendicular lines through a point

Parallel: keep the slope. Perpendicular: flip the slope and change its sign. Then find b from the point.

Write the equation.

1. 89.The line parallel to y = ¹⁄₃x + 3, through (3, 3)
2. 90.The line parallel to y = −³⁄₂x + 3, through (0, 1)
3. 91.The line perpendicular to y = 4x + 3, through (0, −4)
4. 92.The line perpendicular to y = −x − 3, through (3, 4)
5. 93.The line parallel to y = ³⁄₂x − 6, through (−2, 6)
6. 94.The line perpendicular to y = −2x + 6, through (4, −3)

A system is two equations that must both be true. Its solution is the point where the two lines cross.

12.Solving systems

Substitution: when one equation says y = …, put that into the other. Elimination: add or subtract the equations, multiplying first if needed, so that one variable drops out. Parallel lines never cross, so they have no solution; two equations for the same line have infinitely many.

Solve each system.

1. 95.y = 2x + 3 and y = 2x − 1
2. 96.x + y = 5 and 2x + 2y = 10
3. 97.4x + 4y = 12 and 3x + 2y = 10
4. 98.y = −x − 7 and 4x − 2y = 8
5. 99.2x + 3y = 8 and −3x − 2y = −2
6. 100.4x + 2y = 6 and 4x − 2y = −14
7. 101.y = 3x + 3 and 4x − 3y = −14
8. 102.−2x − 2y = −10 and x + 4y = 11
9. 103.y = −2x − 12 and −x + y = 6
10. 104.−3x + 3y = −12 and −x − 3y = −16
11. 105.−2x − y = 16 and −2x − 2y = 20
12. 106.−3x − 2y = −17 and 5x + 4y = 33

13.Systems in words

Name the two unknowns, write two equations, and solve.

Answer each.

1. 107.Plan A costs $20 a month plus $0.10 a minute. Plan B costs $35 a month plus $0.05 a minute. For how many minutes do they cost the same?
2. 108.Tickets cost $5 for children and $8 for adults. 60 tickets sold for $396. How many of each?
3. 109.Two numbers add to 41, and one is 7 more than the other. Find them.
4. 110.A grant fund has $100,000. Large grants are $25,000 and small ones $5,000. It gives 8 grants and spends it all. How many of each?

A function gives exactly one output for each input. f(x) is read “f of x”: the output when the input is x.

14.Finding the output

Put the input in place of x, in parentheses, and work it out.

Find each output.

1. 111.If f(x) = 2x² + 4x + 6, find f(4)
2. 112.If f(x) = 2x + 6, find f(8)
3. 113.If f(x) = 2x − 3, find f(5)
4. 114.If f(x) = 2x + 6, find f(−4)
5. 115.If f(x) = x² − 3x + 6, find f(4)
6. 116.If f(x) = 3x − 3, find f(−3)
7. 117.If f(x) = −x² + 4x − 1, find f(1)
8. 118.If f(x) = x² + x + 9, find f(−3)
9. 119.If f(x) = 4x − 9, find f(7)
10. 120.If f(x) = −3x + 2, find f(−5)
11. 121.If f(x) = 3x² + 3x − 9, find f(−3)
12. 122.If f(x) = −5x − 6, find f(−3)

15.Finding the input

Set the rule equal to the output, then solve for x.

Find x.

1. 123.If f(x) = 5x + 8, find x when f(x) = −2
2. 124.If f(x) = 6x − 3, find x when f(x) = −15
3. 125.If f(x) = −6x, find x when f(x) = 30
4. 126.If f(x) = 6x − 5, find x when f(x) = 37
5. 127.If f(x) = −5x + 7, find x when f(x) = −33
6. 128.If f(x) = −3x − 8, find x when f(x) = −32
7. 129.If f(x) = 7x − 8, find x when f(x) = 41
8. 130.If f(x) = −4x − 6, find x when f(x) = −38

16.Is it a function?

It fails only when one input has two different outputs. Two inputs sharing one output is allowed.

Decide, and give the reason.

1. 131.{(3, 6), (7, −1), (1, 9), (7, −6)}
2. 132.{(0, −6), (4, 9), (8, −3), (7, 1)}
3. 133.{(6, −3), (6, 6), (−2, 1), (0, −1)}
4. 134.{(−5, −6), (−3, −6), (5, 1), (0, 2)}
5. 135.{(0, 5), (−3, −4), (−3, −2), (6, 4)}
6. 136.{(1, −5), (−5, −5), (0, −5), (4, −6)}
7. 137.{(−5, 7), (−1, 4), (0, −6), (−5, −3)}
8. 138.{(−2, 7), (−3, −1), (−1, 3), (1, −6)}

17.The rule from a table

Find how much y changes each time x goes up by 1: that is m. (If x goes up by 2, divide the change in y by 2.) Then find b from any pair.

Write the rule as y = mx + b.

1. 139.Pairs (x, y): (1, 4)   (2, 6)   (3, 8)   (4, 10). Write the rule.
2. 140.Pairs (x, y): (−2, 6)   (−1, 2)   (0, −2)   (1, −6). Write the rule.
3. 141.Pairs (x, y): (−2, 10)   (−1, 6)   (0, 2)   (1, −2). Write the rule.
4. 142.Pairs (x, y): (0, 5)   (1, 11)   (2, 17)   (3, 23). Write the rule.
5. 143.Pairs (x, y): (−2, 0)   (0, 8)   (2, 16)   (4, 24). Write the rule.
6. 144.Pairs (x, y): (−2, −11)   (0, 1)   (2, 13)   (4, 25). Write the rule.