1. Variation
Algebra 2/Trig
Unit 3
Day 1

2. Variation
Relationship between two or more variables using ONLY multiplication or division
Equation uses k (a variation constant) to express the relationship between the two variables

3. Direct Variation 𝑦= 1 5 𝑥 𝑦=3𝑥 𝑘=3 𝑘= 1 5 Equation: 𝑦=𝑘𝑥
Alternative: 𝑘= 𝑦 𝑥
Phrase: x and y vary directly
In an x-y table,
As x increases, y increases
→ absolute value of k is > 1
As x decreases, y decreases
→ absolute value of k is < 1 X Y 1 3 2 6 9 4 12 5 15 X Y 30 6 25 5 15 3 1
𝑦= 1 5 𝑥
𝑘= 1 5
𝑦=3𝑥
𝑘=3

4. Inverse Variation 𝑦= 30 𝑥 𝑘=30 Equation: 𝑦= 𝑘 𝑥 Alternative: 𝑘=𝑦𝑥
Phrase: x and y vary inversely
In an x-y table,
As x increases, y decreases
→absolute value of k is >1
As x decreases, y increases
→absolute value of k is <1
𝑦= 𝑥
= 1 2 ∙ 1 𝑥
𝒚= 𝟏 𝟐𝒙
𝑘= 1 2 X Y 10
0.05 8
0.0625 3
0.1 6 2
0.25 X Y 5 6 4
7.5 3 10 2 15 1 30
𝑦= 30 𝑥
𝑘=30

5. Joint Variation
Direct
Equation: 𝒚=𝒌𝒙𝒛
Phrase: z varies joint directly with x and y
Inverse
Equation: 𝒚= 𝒌 𝒙𝒛
Phrase: z varies joint inversely with x and y
Now that you have notes on direct, inverse, and joint variation, log in to the Socrative to answer a few questions. The questions refer to #1-8 on the top of the next page in your notes.
When you finish the Socrative, come back to the powerpoint slides.

6. Neither Variation – relationships between x and y that use more than just multiplication or division
Linear Equations - 𝑦=𝑚𝑥+𝑏
Quadratic Equations - 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐

7. Determining k values 𝒚= 𝟏 𝟑𝒙 → k = 𝟏 𝟑 𝒚=𝟓𝒙→ k = 5
Now that you know the type of variation for each example #1-8, state the k value for the variation.
Just as a reminder:
𝒚= 𝟏 𝟑𝒙 → k = 𝟏 𝟑
𝒚=𝟓𝒙→ k = 5

8. Want to stop here and try the examples on your own?
Yes – close the powerpoint, log off your computer and return it to the cart.
No – keep clicking through the slides to see the examples and copy them into your notes

9. Example 9
Calculate the value of k. Write the equation, and then solve for y given a value of x.
a. The variables x and y vary inversely, when x = 5 then y = 8.
Inverse equation → 𝑦= 𝑘 𝑥
Sub in values given → 8= 𝑘 5
Solve for k → 5∙8= 𝑘 5 ∙5
40=𝑘
Write the equation → 𝑦= 40 𝑥
b. Find the value of y when x = -2.
𝑦= 40 𝑥
𝑦= 40 −2
𝑦=−20

10. Examples 10, 11, 12 Determine the type and equation given the table.
Complete the columns for 𝑦 𝑥 and 𝑦∙𝑥.
Use the information above to decide if the variation is direct, inverse, or neither.
If it is direct or inverse, the value you found that was the same for every row is the k value in the equation.
DIRECT VARIATION
𝑦 𝑥 = the same number for every row of the table
INVERSE VARIATION
𝑦∙𝑥 = the same number for every row of the table

11. All done! Close the powerpoint Log off Return the laptop to the cart
Pick up a practice worksheet and continue with those problems