Math Journal 9-11 Find the value of the function 𝑓 𝑥 , 𝑔 𝑥 , ℎ 𝑥 when the input is 𝑥=2 𝑎𝑛𝑑 𝑥=−1 𝑓 𝑥 =𝑥 +10 2. 𝑔 𝑥 = 2 𝑥 3. ℎ 𝑥 = 5𝑥 𝑓 2 = 𝑔 2 = ℎ 2 = 𝑓 −1 = 𝑔 −1 = ℎ −1 =

Unit 2 Day 5: Comparing Functions Essential Questions: How do linear, exponential, absolute value, and square root functions differ from each other? What causes a shift in a graph?

Vocabulary Linear: a straight line with a constant rate of change. Exponential: a function whose rate of change increases/decreases over time. Square Root: a number that produces a specific quantity when multiplied by itself. Ex: Intercepts (x & y): the points where the graph of a function crosses the x and / or y axis.

General Rule For Graphing This rule always works: when in doubt make a table and plot it out! x y = 2x y -2 y = 2(-2) -4 -1 y = 2(-1) y = 2(0) 1 y = 2(1) 2 y = 2(2) 4 Plug whatever x-values you want into the given function. Make it easy on yourself by choosing easy numbers!

Linear Functions f(x) = 10x 20 30 40 50 1 2 3 4 5 f(x) = 10x This function can represent getting paid 10 dollars per hour. The variable (x) represents hours. Notice how it has a constant rate of change of $10 for every 1 hour.

Linear Functions: Shifts 10 20 30 40 50 1 2 3 4 5 f(x) = 10x + 15 This function can represent getting paid 10 dollars per hour. The variable (x) represents hours. What could the “+15” stand for?

Linear Functions: Shifts 10 20 30 40 50 1 2 3 4 5 f(x) = 10x - 20 This function can represent getting paid 10 dollars per hour. The variable (x) represents hours. What could the “- 20” stand for?

Example 1: Match the function to its graph. f(x) = x - 2 f(x) = x + 2 f(x) = x f(x) = x - 5

Example 1: Match the function to its graph. f(x) = x f(x) = x - 5 f(x) = x + 2 f(x) = x - 2

Exponential Functions 2 𝑥 3 𝑥 10 𝑥 (-1, 1/2 ) (-1, 1/3) (-1, 1/10) (0,1 ) (0,1) (1,2) (1, 3) (1,10) (2,4) (2, 9) (2, 100) (3,8) (3, 27) (3, 1000) Why do they all have the same y-intercept? Why do you think these graphs are growing at a different rate?

Absolute Value Functions An absolute value graph always has a “V” shape. f(x) = |x| f(x) = -|x| X y = |x| y -2 y = |-2| 2 -1 y = |-1| 1 y = |0| y = |1| y = |2| X y = -|x| y -2 y = -|-2| 2 -1 y = -|-1| 1 y = -|0| y = -|1| y = -|2|

Square Root Functions Square root functions are curved and usually level off as ‘x’ gets bigger. Why do you think it levels off? Why isn’t the negative part of the graph being used?

Square Root Functions: Shifts Square root functions shift the same as other functions: Subtracting a number causes a shift down Adding a number causes a shift up

Example 2: Matching f(x) = 2x f(x) = |x| f(x) = x f(x) = ✓x

Example 2: Matching f(x) = ✓x f(x) = x f(x) = |x| f(x) = 2x

Example 3: Matching 𝑓 𝑥 = 𝑥 +3 f(x) = 2x - 3 f(x) = |x| - 3 𝑓 𝑥 = 𝑥 +3 f(x) = 2x - 3 f(x) = |x| - 3 f(x) = x + 2 -5

Example 3: Matching 𝑓 𝑥 = 𝑥 +3 f(x) = 2x - 3 f(x) = x + 2 𝑓 𝑥 = 𝑥 +3 f(x) = x + 2 -5 f(x) = |x| - 3 f(x) = 2x - 3

Test Your Skills! http://www.regentsprep.org/Regents/math/algtrig/AT P5/FuncPrac.htm

Summary Essential Questions: How do linear, exponential, absolute value, and square root functions differ from each other? What causes a shift in a graph? Take 1 minute to write 2 sentences answering the essential questions.