Linear Interpolation worksheet

1. x: 24

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{60-52}{24-10}$$=\frac{8}{14}$$=0.571429$

2. Rearrange the line for x: start at x₁ and move by the change in y divided by the slope

   $\text{x = x₁ + (y − y₁) × (x₂ − x₁) / (y₂ − y₁)}$

3. Put in the target y and the two points to get x

   $x$$=10+\left(60-52\right)\times \frac{24-10}{60-52}$$=24$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{24-10}{24-10}$$=1$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=52-0.571429\times 10$$=46.285714$

2. Interpolated y: 53.46

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{100-45}{22-9}$$=\frac{55}{13}$$=4.230769$

2. Use the interpolation formula: start at y₁ and add the slope times how far x is from x₁

   $\text{y = y₁ + (x − x₁) × (y₂ − y₁) / (x₂ − x₁)}$

3. Put in the target x and the two points to get y

   $y$$=45+\left(11-9\right)\times \frac{100-45}{22-9}$$=53.461538$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{11-9}{22-9}$$=0.153846$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=45-4.230769\times 9$$=6.923077$

3. Interpolated y: 97.14

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{110-50}{24-10}$$=\frac{60}{14}$$=4.285714$

2. Use the interpolation formula: start at y₁ and add the slope times how far x is from x₁

   $\text{y = y₁ + (x − x₁) × (y₂ − y₁) / (x₂ − x₁)}$

3. Put in the target x and the two points to get y

   $y$$=50+\left(21-10\right)\times \frac{110-50}{24-10}$$=97.142857$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{21-10}{24-10}$$=0.785714$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=50-4.285714\times 10$$=7.142857$

4. Interpolated y: 57.25

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{64-55}{24-12}$$=\frac{9}{12}$$=0.75$

2. Use the interpolation formula: start at y₁ and add the slope times how far x is from x₁

   $\text{y = y₁ + (x − x₁) × (y₂ − y₁) / (x₂ − x₁)}$

3. Put in the target x and the two points to get y

   $y$$=55+\left(15-12\right)\times \frac{64-55}{24-12}$$=57.25$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{15-12}{24-12}$$=0.25$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=55-0.75\times 12$$=46$

5. x: 13.16

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{70-45}{17-9}$$=\frac{25}{8}$$=3.125$

2. Rearrange the line for x: start at x₁ and move by the change in y divided by the slope

   $\text{x = x₁ + (y − y₁) × (x₂ − x₁) / (y₂ − y₁)}$

3. Put in the target y and the two points to get x

   $x$$=9+\left(58-45\right)\times \frac{17-9}{70-45}$$=13.16$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{13.16-9}{17-9}$$=0.52$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=45-3.125\times 9$$=16.875$

6. x: 15.26

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{90-28}{23-13}$$=\frac{62}{10}$$=6.2$

2. Rearrange the line for x: start at x₁ and move by the change in y divided by the slope

   $\text{x = x₁ + (y − y₁) × (x₂ − x₁) / (y₂ − y₁)}$

3. Put in the target y and the two points to get x

   $x$$=13+\left(42-28\right)\times \frac{23-13}{90-28}$$=15.258065$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{15.258065-13}{23-13}$$=0.225806$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=28-6.2\times 13$$=\mathrm{-52.6}$

7. x: 8

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{100-64}{16-10}$$=\frac{36}{6}$$=6$

2. Rearrange the line for x: start at x₁ and move by the change in y divided by the slope

   $\text{x = x₁ + (y − y₁) × (x₂ − x₁) / (y₂ − y₁)}$

3. Put in the target y and the two points to get x

   $x$$=10+\left(52-64\right)\times \frac{16-10}{100-64}$$=8$

4. The target lies outside the two known points, so this is extrapolation: the straight line is extended beyond the data

   $t$$=\frac{8-10}{16-10}$$=\mathrm{-0.333333}$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=64-6\times 10$$=4$

8. Interpolated y: 54.36

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{65-52}{20-9}$$=\frac{13}{11}$$=1.181818$

2. Use the interpolation formula: start at y₁ and add the slope times how far x is from x₁

   $\text{y = y₁ + (x − x₁) × (y₂ − y₁) / (x₂ − x₁)}$

3. Put in the target x and the two points to get y

   $y$$=52+\left(11-9\right)\times \frac{65-52}{20-9}$$=54.363636$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{11-9}{20-9}$$=0.181818$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=52-1.181818\times 9$$=41.363636$

9. x: 10.92

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{85-60}{17-9}$$=\frac{25}{8}$$=3.125$

2. Rearrange the line for x: start at x₁ and move by the change in y divided by the slope

   $\text{x = x₁ + (y − y₁) × (x₂ − x₁) / (y₂ − y₁)}$

3. Put in the target y and the two points to get x

   $x$$=9+\left(66-60\right)\times \frac{17-9}{85-60}$$=10.92$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{10.92-9}{17-9}$$=0.24$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=60-3.125\times 9$$=31.875$

10. Interpolated y: 52.5

1. Find the slope of the line through the two points: the change in y divided by the change in x

   $m$$=\frac{80-45}{23-9}$$=\frac{35}{14}$$=2.5$

2. Use the interpolation formula: start at y₁ and add the slope times how far x is from x₁

   $\text{y = y₁ + (x − x₁) × (y₂ − y₁) / (x₂ − x₁)}$

3. Put in the target x and the two points to get y

   $y$$=45+\left(12-9\right)\times \frac{80-45}{23-9}$$=52.5$

4. The target lies between the two known points, so this is interpolation

   $t$$=\frac{12-9}{23-9}$$=0.214286$

5. Write the line as y = mx + b, where b is where it crosses the y-axis: b = y₁ − m·x₁

   $b$$=45-2.5\times 9$$=22.5$