FUNCTION FAMILIES TOOLKIT y = xy = x2 y = x3 y ... AWS
NAME: ______________________________
FUNCTION FAMILIES TOOLKIT In Precalculus, you developed Function Families. This Toolkit will be the basis of all our work in AP Calculus. You will learn how to extend the ideas in your Toolkit much, much further. You will need multiple sheets of the Functions Families Toolkit Page. You will want to keep & refer to your function families mini project throughout the year, so be sure to make it neat, organized, and easy to find & use.
PART 1: GENERAL FUNCTION FAMILIES & INVERSES 1) With your group, make graphs or careful sketches of each of the parent equations below. y=x y= x x 2 + y2 = 1
y = x2 y = x3
y = ex y = sin x
2) Label each equation with the name of its family. Absolute value Circle Rational Linear
1 x
y=
Cubic Quadratic
Exponential Sinusoidal
3) Write the general equation for each parent. For example, for the sinusoidal, it’s y a sin(bx c) d . Give explanations for what a, b, c and d tell you about the graphs. 4) If any of these equations are not functions, say so next to its graph. If any have asymptotes, label them and note it. 5) Write the domain and range next to each graph. 6) Describe the symmetry of the graph, if any (using the words and explaining the meaning of “even” and “odd” functions). 7) Make sure you are putting scales on your graphs and labeling all x- and y-intercepts. 8) INVERSES - Use the toolkit page specifically for inverses for this portion of the project. On each graph, you will sketch both the parent graph and the inverse. You will write both functions on the toolkit page. Again, I want you to use the eight basic functions above. For some of these (you’ll have to figure out which ones), the inverse is the SAME as the original function. Therefore, in order to show up on your graph, you should NOT use the parent, but a very simple transformation of the parent. For example, do not use y = x ; use y = x + 3 or something like that.
PART 2: INVESTIGATING POLYNOMIAL FUNCTIONS Some of your parents have similar equations. For example, y = x 2 and y = x 3 both look like y = x n . You might recall from Precalculus that all the functions that are based on y = x n are called Polynomial Functions. Investigate polynomial functions & answer the following questions, with pictures, on another toolkit page. You will add this page to your Function Families Toolkit. 1) As n gets bigger (but stays a positive whole number), what changes in the graphs? Give several examples, and write a summary statement that describes any patterns you find. 2) As n changes, what stays the same?
3) Now try n’s that are fractions but still positive. What happens to the graph? Give several examples, and write a summary statement that describes any patterns you find. 4) How about negative values for n. What happens then? Give several examples, and write a summary statement that describes any patterns you find.
PART 3: MORE ABOUT TRIG FUNCTIONS Here’s a quick reminder about the six trigonometric functions you’ve studied over the last three years:
FUNCTION
HOW TO SAY IT LIKE A PRO
HOW TO FIND IT
𝑦 = sin 𝑥
Sine
--
𝑦 = cos 𝑥
Cosine
--
𝑦 = tan 𝑥
Tangent
tan x =
sin x cos x
𝑦 = cot 𝑥
Cotangent (like “co-pilot”)
cot x =
cos x sin x
𝑦 = sec 𝑥
Secant (“see can’t”)
sec x =
1 cos x
𝑦 = csc 𝑥
Cosecant
csc x =
1 sin x
Because the last four trig functions are based on sine and cosine, their graphs are related to the graphs of y = sin x and y = cos x as well as to each other. Your job is to investigate these relationships & add a new page to your Function Families Toolkit. 1) Get a new toolkit sheet. Use the front side for the first three trig functions & the back side for the second three functions. For each graph, label the basics – x-intercepts, y-intercepts, etc. and state what the period and amplitude is (if appropriate). Look for and describe the type of symmetry (ODD or EVEN) each graph has. 2) Make accurate graphs for each function. All the graphs except for sine and cosine should have asymptotes – be sure to mark them & label the x values where they occur! 3) Look at the asymptotes for those four functions. Why do the asymptotes occur where they do? Your group should come up with a clear, concise mathematical explanation & include it in your Toolkit. 4) Look at the x- and y-intercepts. Why do the intercepts occur where they do? Come up with a clear, concise mathematical explanation & include it in your Toolkit. Remember, your toolkit page should “stand alone.” It should fully show & explain all the relationships & reasons you found. Use color, arrows, small diagrams, and other communication tools to help you explain. Be sure to include whatever reasons you found to justify every claim you make.
PART 4: FUNCTION TRANSFORMATIONS Understanding how basic transformations change the graphs of function is a SUPER important part of being successful in calculus. Add another page to your Function Families Toolkit titled FUNCTION TRANSFORMATIONS. Your page should explain the effect of carrying out the following basic transformations on the parent graph y = x 2
Adding or subtracting a number from the x value Adding or subtracting a number from the entire function Multiplying a positive number greater than 1 by the x value (use parabolas and lines as examples) Multiplying a positive number less than 1 by the x value Multiplying the x value by -1. (NOTE: Use y = (x - 3)2 +1, not y = x 2 ) Multiplying the entire function by -1.
Function Transformations
Blah blah Blah blah Blah blah
Blah blah
For each, be sure to… Describe the effect in words Show the effect with an example (a graph of the original function and a graph of the transformed function in a different color) Use color, arrows, small diagrams, and other communication tools to help you explain. Remember, your toolkit page should “stand alone.” It should fully show & explain the effects of each transformation. Be sure to include whatever reasons you found to justify every claim you make. You may use ANY and ALL resources you can find (textbook, internet, graphing calculator, old pre-calc notes, friends, family, etc.). If you are unsure where to start, choose a function, and use your graphing calculator or Desmos to graph the function with different transformations applied.
PART 5: THE UNIT CIRCLE Knowing the unit circle is another HUGE part of being successful in calculus. Reconstruct the Unit Circle below using ANY and ALL resources you can find (textbook, internet, graphing calculator, old precalc notes, friends, family, etc.). Use exact values, not decimal approximations. Before your Unit Quiz next week, add this information to a page in your Function Families Toolkit. You may simply fill out the circle and the table below and staple this to your toolkit. Label the circle with the trig values of sinθ and cosθ
Degrees
Radians
x (cosine)
y (sine)
Degrees
0
180
30
210
45
225
60
240
90
270
120
300
135
315
150
330 360
Radians
x (cosine)
y (sine)
FUNCTION FAMILIES TOOLKIT In Precalculus, you developed Function Families. This Toolkit will be the basis of all our work in AP Calculus. You will learn how to extend the ideas in your Toolkit much, much further. You will need multiple sheets of the Functions Families Toolkit Page. You will want to keep & refer to your function families mini project throughout the year, so be sure to make it neat, organized, and easy to find & use.
PART 1: GENERAL FUNCTION FAMILIES & INVERSES 1) With your group, make graphs or careful sketches of each of the parent equations below. y=x y= x x 2 + y2 = 1
y = x2 y = x3
y = ex y = sin x
2) Label each equation with the name of its family. Absolute value Circle Rational Linear
1 x
y=
Cubic Quadratic
Exponential Sinusoidal
3) Write the general equation for each parent. For example, for the sinusoidal, it’s y a sin(bx c) d . Give explanations for what a, b, c and d tell you about the graphs. 4) If any of these equations are not functions, say so next to its graph. If any have asymptotes, label them and note it. 5) Write the domain and range next to each graph. 6) Describe the symmetry of the graph, if any (using the words and explaining the meaning of “even” and “odd” functions). 7) Make sure you are putting scales on your graphs and labeling all x- and y-intercepts. 8) INVERSES - Use the toolkit page specifically for inverses for this portion of the project. On each graph, you will sketch both the parent graph and the inverse. You will write both functions on the toolkit page. Again, I want you to use the eight basic functions above. For some of these (you’ll have to figure out which ones), the inverse is the SAME as the original function. Therefore, in order to show up on your graph, you should NOT use the parent, but a very simple transformation of the parent. For example, do not use y = x ; use y = x + 3 or something like that.
PART 2: INVESTIGATING POLYNOMIAL FUNCTIONS Some of your parents have similar equations. For example, y = x 2 and y = x 3 both look like y = x n . You might recall from Precalculus that all the functions that are based on y = x n are called Polynomial Functions. Investigate polynomial functions & answer the following questions, with pictures, on another toolkit page. You will add this page to your Function Families Toolkit. 1) As n gets bigger (but stays a positive whole number), what changes in the graphs? Give several examples, and write a summary statement that describes any patterns you find. 2) As n changes, what stays the same?
3) Now try n’s that are fractions but still positive. What happens to the graph? Give several examples, and write a summary statement that describes any patterns you find. 4) How about negative values for n. What happens then? Give several examples, and write a summary statement that describes any patterns you find.
PART 3: MORE ABOUT TRIG FUNCTIONS Here’s a quick reminder about the six trigonometric functions you’ve studied over the last three years:
FUNCTION
HOW TO SAY IT LIKE A PRO
HOW TO FIND IT
𝑦 = sin 𝑥
Sine
--
𝑦 = cos 𝑥
Cosine
--
𝑦 = tan 𝑥
Tangent
tan x =
sin x cos x
𝑦 = cot 𝑥
Cotangent (like “co-pilot”)
cot x =
cos x sin x
𝑦 = sec 𝑥
Secant (“see can’t”)
sec x =
1 cos x
𝑦 = csc 𝑥
Cosecant
csc x =
1 sin x
Because the last four trig functions are based on sine and cosine, their graphs are related to the graphs of y = sin x and y = cos x as well as to each other. Your job is to investigate these relationships & add a new page to your Function Families Toolkit. 1) Get a new toolkit sheet. Use the front side for the first three trig functions & the back side for the second three functions. For each graph, label the basics – x-intercepts, y-intercepts, etc. and state what the period and amplitude is (if appropriate). Look for and describe the type of symmetry (ODD or EVEN) each graph has. 2) Make accurate graphs for each function. All the graphs except for sine and cosine should have asymptotes – be sure to mark them & label the x values where they occur! 3) Look at the asymptotes for those four functions. Why do the asymptotes occur where they do? Your group should come up with a clear, concise mathematical explanation & include it in your Toolkit. 4) Look at the x- and y-intercepts. Why do the intercepts occur where they do? Come up with a clear, concise mathematical explanation & include it in your Toolkit. Remember, your toolkit page should “stand alone.” It should fully show & explain all the relationships & reasons you found. Use color, arrows, small diagrams, and other communication tools to help you explain. Be sure to include whatever reasons you found to justify every claim you make.
PART 4: FUNCTION TRANSFORMATIONS Understanding how basic transformations change the graphs of function is a SUPER important part of being successful in calculus. Add another page to your Function Families Toolkit titled FUNCTION TRANSFORMATIONS. Your page should explain the effect of carrying out the following basic transformations on the parent graph y = x 2
Adding or subtracting a number from the x value Adding or subtracting a number from the entire function Multiplying a positive number greater than 1 by the x value (use parabolas and lines as examples) Multiplying a positive number less than 1 by the x value Multiplying the x value by -1. (NOTE: Use y = (x - 3)2 +1, not y = x 2 ) Multiplying the entire function by -1.
Function Transformations
Blah blah Blah blah Blah blah
Blah blah
For each, be sure to… Describe the effect in words Show the effect with an example (a graph of the original function and a graph of the transformed function in a different color) Use color, arrows, small diagrams, and other communication tools to help you explain. Remember, your toolkit page should “stand alone.” It should fully show & explain the effects of each transformation. Be sure to include whatever reasons you found to justify every claim you make. You may use ANY and ALL resources you can find (textbook, internet, graphing calculator, old pre-calc notes, friends, family, etc.). If you are unsure where to start, choose a function, and use your graphing calculator or Desmos to graph the function with different transformations applied.
PART 5: THE UNIT CIRCLE Knowing the unit circle is another HUGE part of being successful in calculus. Reconstruct the Unit Circle below using ANY and ALL resources you can find (textbook, internet, graphing calculator, old precalc notes, friends, family, etc.). Use exact values, not decimal approximations. Before your Unit Quiz next week, add this information to a page in your Function Families Toolkit. You may simply fill out the circle and the table below and staple this to your toolkit. Label the circle with the trig values of sinθ and cosθ
Degrees
Radians
x (cosine)
y (sine)
Degrees
0
180
30
210
45
225
60
240
90
270
120
300
135
315
150
330 360
Radians
x (cosine)
y (sine)
