# LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

Size: px
Start display at page:

Download "LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II"

Transcription

1 LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also, for a function, f(), the inverse function is written as Eercise #: Find (a) f () = -3 + f 1 ( ). (b) f 1 ( ). 1 f ( ) 5 1 The inverses of polnomial and rational functions can also be found. Eercise #3: Find formulas for the inverse of each of the following simple rational or polnomial functions. 3 (a) g ( ) (b) h ( ) (c) 3 ( ) (d) f( ) (e) 4 (f) f 5 ( ) ( 4) 6 Modified from emathinstruction, RED HOOK, NY 1571, 015

2 In the net lesson, we will be working on transforming functions. For this lesson, we will practice writing new functions that are related to these transformations. Eercise #4: The following table includes two parent functions and a quadratic function. Complete each of the following function changes. You do not need to simplif the resulting function. Do not do the last column. Each function change is written as if the function were g(). Make the same change to m() and h(). Function Change f(-) A. Cubic Parent Function g( ) 3 B. Square Root Parent Function m( ) C. Quadratic Function h ( ) 4 3 Transformation -f() f(+) f()- f(-)+5 3f() f() ***** 1 ( ) f 1 f 3 f( 3) f( ) 5 ***** ***** ***** Modified from emathinstruction, RED HOOK, NY 1571, 015

3 3 LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II HOMEWORK 1. Find formulas for the inverse of each of the following functions. (a) 5 (b) 3 4 (c) f () = (d) g() = -( +1) 3 + (e) h() = 3-3 (f) g() = 1-1 (g) 5 ( 4) 5 (h) 4 3 Modified from emathinstruction, RED HOOK, NY 1571, 015

4 . Complete the following table. Column A and B are parent functions, and Column C is a cubic function. NOTE: Each function change is written as if the function were f(). Make the change same change to j() and g(). You do not need to simplif. 4 Function Change A. Absolute Value Parent Function f ( ) B. Quartic Parent Function j( ) 4 C. Cubic Function g 3 ( ) f(-) -f() f(-4) f()+1 f(+3)- f() f(3) 1 ( ) 3 f f 1 4 f( 1) f( ) 6 Modified from emathinstruction, RED HOOK, NY 1571, 015

5 5 LESSON #43 FUNCTION TRANSFORMATIONS COMMON CORE ALGEBRA II In the previous lesson, we worked with function changes such as f(-) and f(+3). Each of these changes causes a predictable change in the graph of the function known as a transformation. For eample, making the negative, f(-), will alwas cause the graph to change in the same wa. The easiest function to see all of the transformations is the square root parent function. Eercise #1: Go back to lesson #7, graph the square root parent function as Y1 and each of the changes (transformations) as Y. Determine the transformation that occurred for each problem, and write it in the last column. Eercise #: Summarize our findings in the table below. Function Notation Transformation Groups of Transformations f(-) -f() f(+a) f(-a) f()+a f()-a af() where a 1 af() where 0 a 1 f(a) where a 1 f(a) where 0 a 1 Note: We will be working with horizontal stretches and compressions more etensivel in the net unit. Modified from emathinstruction, RED HOOK, NY 1571, 015

6 6 Eercise #3: What is the equation of the absolute value parent function, a()=, after each of the following transformations. a. Shift Right 3 b. Reflection in the -ais. c. Shift Down d. Vertical Stretch of 3. e. Reflection in the -ais f. Shift left 3, up 4. g. Vertical Compression of 1. Eercise #4: The function f function, g, is defined b g f is shown on the grid below. A second 3 1. (a) Identif how the graph of f has been transformed to produce the graph of g and sketch it on the grid. (b) A third function, h, is defined b h()= f (). Identif how the graph of f has been transformed to produce the graph of h and sketch it on the grid. (Note: The maimum of h will be off the grid). Eercise #5: If the parabola = were shifted 6 units left and units down, its resulting equation would be which of the following? Verif b graphing our answer and seeing if its turning point is at (-6,-). (1) = ( + 6) + (3) = ( - 6) - () = ( + 6) - (4) = ( - 6) + Modified from emathinstruction, RED HOOK, NY 1571, 015

7 7 Eercise #6: The graph of a function (b) the graph of f. f is shown below on two grids. Sketch (a) the graph of f and (a) Graph and label. (b) Graph and label Eercise #7: Determine an equation for the linear function g 5 7 -ais. Label our equations. both after a reflection in the -ais and Eercise #8: If the point 3,1 lies on the graph of the function f, which of the following points must lie on the graph of 3 f? (1) 9, 36 (3) 3, 4 () 3, 36 (4) 9,1 Eercise #9: If f 5 3 and is which of the following? g is the reflection of f across the -ais, then an equation of g (1) g 5 3 (3) g 5 3 () g 5 3 (4) g 5 3 Modified from emathinstruction, RED HOOK, NY 1571, 015

8 8 LESSON #43 - FUNCTION TRANSFORMATIONS COMMON CORE ALGEBRA II HOMEWORK 1. What is the equation of the quintic parent function, f ()= 5, after each of the following transformations? You do not need to simplif. a. Shift left 4 b. Reflection in the -ais c. Shift Up 1 d. Vertical Compression of 1 4 e. Reflection in the -ais f. Shift right 5, down g. Vertical Stretch of 3. Which of the following represents the turning point of f ()=( -8) -4? (1) (-8,-4) (3) (-8,4) () (8,4) (4) (8,-4) 3. Consider the quadratic function f ()= The quadratic functions g and h are defined b the formula, g()= f () and h()= 1 f (). Determine formulas for both g and h in standard form. 4. Which of the following equations would represent the graph of the parabola reflection in the -ais? after a (1) () (3) (4) Modified from emathinstruction, RED HOOK, NY 1571, 015

9 5. If the point 3, 5 lies on the graph of a function h then which of the following points must lie on the graph of the function h? (1) 3, 5 (3) 5, 3 () 3, 5 (4) 3, 5 6. If the point 6,10 lies on the graph of f graph of 1 f? (1) 3, 5 (3) 6, 5 () 3,10 (4) 1, 0 7. If the quadratic function f has a turning point at 3, 7 defined b g f (1) 7,1 (3) 4, 5 () 1,1 (4) 4, 5 8. The graph of 4 5 have a turning point? then which of the following points must lie on the then where does the quadratic function g f 4 is show below on two separate grids. Give an equation and sketch a graph for the functions (a) f and (b) f. 9 (a) (b) Modified from emathinstruction, RED HOOK, NY 1571, 015

10 10 9. Given the function f shown graphed on the grid, create a graph for each of the following functions and label on the grid. (a) g f (b) h f 3 (c) k ( ) = 1 f () 10. The graph of the function f is shown on the grid below. The function g is defined b the formula f g 3 1. (a) Graph and label g on the aes. (b) What is the smallest solution to the equation f g? (c) If h g 3, eplain wh the equation h f has no solutions. Modified from emathinstruction, RED HOOK, NY 1571, 015

11 11 LESSON #44 MULTIPLE TRANSFORMATIONS COMMON CORE ALGEBRA II The following activit will help ou start to think about multiple transformations. Eercise #1: Identif the parent function for each transformation. Draw arrows to each transformation that has happened to the parent function and identif them. a. g( ) 3 7 b. 3 7 c. h( ) d. r()=5( -3) -9 Eercise #: Consider the function, g()= (a) Graph the function =g() on the grid shown. (b) Describe the transformations that have occurred to the graph of = to produce the graph of =g(). Specif the transformations and the order. Eercise #3: How would the graph of the function h() compare to the graph of f() if h is defined b the formula h()= - f (-)? Modified from emathinstruction, RED HOOK, NY 1571, 015

12 1 Eercise #4: Which of the following equations represents the graph shown below? (1) 3 4 (3) 3 4 () 3 4 (4) 3 4 g Eercise #5: Consider the function 4. What two transformations have occurred to the graph of to produce the graph of g? Specif the transformations and the order in which the occurred. Note that there eists more than one correct answer. Graph on our calculator to verif. Eercise #6: If the quadratic function function h g 3 5? (1) It has a -intercept of -5. () It has a -intercept of 1. (3) It has a -intercept of -15. (4) It has a -intercept of 31. g has a -intercept of 1, which of the following is true about the Eercise #7: The graph of 1 3. h is shown below. The function f is defined b f h (a) What three transformations have occurred to the graph of h to produce the graph of f? Specif the transformations and the order the occurred in. (b) Graph and label the function that contains h. f on the grid below Modified from emathinstruction, RED HOOK, NY 1571, 015

13 Eercise #8: The function 3 f h h has a range given b the interval,10. The function f 8. Which of the following gives the range of (1) 11, 3 (3) 15, 7 () 8,1 (4) 6, 3 f? 13 is defined b Modified from emathinstruction, RED HOOK, NY 1571, 015

14 14 LESSON #44 - MULTIPLE TRANSFORMATIONS COMMON CORE ALGEBRA II HOMEWORK 1. For each of the following functions, identif the parent function. In the last column, identif the transformations that occurred on the parent function and their order. Modified from emathinstruction, RED HOOK, NY 1571, 015

15 . If the function h is defined as vertical stretch b a factor of followed b a reflection in the -ais of the function f then h 1 f (1) f (3) 1 () f (4) f If the graph of is compressed b a factor of 1 3 in the -direction and then shifted 4 units down, the resulting graph would have an equation of (1) (3) 4 3 () 3 4 (4) The quadratic function f has a turning point at 3,6. The quadratic f turning point of (1), 9 (3) 3, 7 () 1, 7 (4) 1, would have a 5. The graph of f is shown below. Consider the function g defined b g f 3. (a) What two transformations have occurred to the graph of f in order to produce the graph of g? Specif both the transformations and their order. (b) Graph and label g Modified from emathinstruction, RED HOOK, NY 1571, 015

16 16 LESSON #45 - EVEN AND ODD FUNCTIONS COMMON CORE ALGEBRA II Recall that functions are simpl rules that convert inputs into outputs. These rules then get placed into various categories, such as linear functions, eponential functions, quadratic functions, etcetera, based on shared characteristics. In this lesson ou will learn another wa to classif some functions that have useful smmetries. EVEN AND ODD FUNCTIONS A function is known as even if for ever value of in the domain of. A function is known as odd if ever value of in the domain of. The terms even function and odd function come from the properties of power functions that have an even eponent or an odd eponent, respectivel. Eercise #1: Draw the following sketches. Identif the tpe of smmetr for the graph and eplain how this makes sense based on the definition above. (a) Draw a basic sketch of an even power function in the form, where a and b are positive. (b) Draw a basic sketch of an odd power function in the form, where a and b are positive. Modified from emathinstruction, RED HOOK, NY 1571, 015

17 Eercise #3: Consider the partial graph of the function f function if in (a) (a) even f is even and in (b) f 17 shown twice below. Sketch the other half of the is odd. The three coordinate pairs are listed to help ou plot. (b) odd 0, 0, 3, 5, 7, 0, 0, 3, 5, 7, Eercise #4: g 4. Eercise #5: Let's investigate 3 (a) Use our calculator's table option to fill in the following table. What tpe of function is this? Eplain g (b) Sketch a graph of the window indicated. g using our calculator and Modified from emathinstruction, RED HOOK, NY 1571, 015

18 Determine if the functions on the left side are Even, Odd, or neither. Place a check in the appropriate column. At the bottom of the sheet, use the total number of checks in each column to see if the stated equation is true. Check our work if the epression does not equal. Use a table, graph, or algebraic work. 18 COMPLETE THIS PAGE AS A PART OF YOUR HOMEWORK. Modified from emathinstruction, RED HOOK, NY 1571, 015

19 19 FLUENCY LESSON #45 - EVEN AND ODD FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK 1. Complete the previous page.. Given the partiall filled out table below for f, fill out the rest of it based on the function tpe. (a) Even (b) Odd Half of the graph of f is shown below. Sketch the other half based on the function tpe. (a) Even (b) Odd Modified from emathinstruction, RED HOOK, NY 1571, 015

20 0 5. If f is an even function and f 3 5 then what is the value of 4 f 3 f 3? (1) 30 (3) 10 () 60 (4) 6 6. Which of the following functions is even? Show algebraicall how ou arrived at our choice. (1) 4 (3) 9 () 6 (4) 4 7. The function f 4 is either even or odd. Determine which b eploring the function using tables on our calculator. Cop the table. Show algebraicall that the choice is correct. REASONING 8. Even functions have smmetr across the -ais. Odd functions have smmetr across the origin. Can a function have smmetr across the -ais? Wh or wh not? Modified from emathinstruction, RED HOOK, NY 1571, 015

21 1 LESSON #46 - CIRCLES AND SYSTEMS COMMON CORE ALGEBRA II We will start this lesson b reviewing part of what ou learned about circles in common core geometr. THE EQUATION OF A CIRCLE A circle whose center is at and whose radius is r is given b: Eercise #1: Which of the following equations would have a center of 3, 6 and a radius of 3? (1) (3) () (4) Eercise #: For each of the following equations of circles, determine both the circle s center and its radius. If its radius is not an integer, epress it in simplest radical form and rounded to the nearest tenth. (a) (b) (c) 11 (d) 1 1 (e) 3 49 (f) (g) 64 (h) 4 0 (i) 57 Eercise #3: Based on our answers, graph circles d, g, and h on the grid below. Label each circle. Modified from emathinstruction, RED HOOK, NY 1571, 015

22 In Unit # ou learned how to solve sstems of equations involving a quadratic equation and a linear equation. Now we will work with sstems that involve a linear equation and a circle. These can be solved both algebraicall and graphicall. 1 Eercise #4: Solve the following sstem of equations both algebraicall and graphicall. 5 Algebraicall: Graphicall: 7 Eercise #5: Solve the following sstem of equations both algebraicall and graphicall: ( ) ( ) 9 Algebraicall: Graphicall: Modified from emathinstruction, RED HOOK, NY 1571, 015

23 3 5 Eercise #6: Solve the following sstem of equations both algebraicall and graphicall: 97 Algebraicall: Graphicall: Note: It can be difficult to find the eact points of intersection if the circle is not graphed perfectl. An algebraic method is recommended. Modified from emathinstruction, RED HOOK, NY 1571, 015

24 4 FLUENCY LESSON #46 - CIRCLES AND SYSTEMS COMMON CORE ALGEBRA II HOMEWORK 1. Each of the following is an equation of a circle. State the circle s center and radius. If its radius is not an integer, epress it in simplest radical form and rounded to the nearest tenth. (a) 50 (b) (c) (d) (e) 1 (f) Which of the following is true about a circle whose equation is (1) It has a center of 5, 3 and an area of 1. () It has a center of 5, 3 and a diameter of 6. (3) It has a center of 5, 3 and an area of 36. (4) It has a center of 5, 3 and a circumference of ? 3. Which of the following represents the equation of the circle shown graphed below? (1) 3 16 () 3 4 (3) 3 4 (4) 3 16 Modified from emathinstruction, RED HOOK, NY 1571, 015

25 Solve the following sstem of equations both algebraicall and graphicall. 58 Algebraicall: Graphicall: 8. Find the intersection of the circle ( 3) ( 4) 9 and 4 both algebraicall and graphicall. Algebraicall: Graphicall: Modified from emathinstruction, RED HOOK, NY 1571, 015

26 6 LESSON #47 - SYSTEMS OF LINEAR EQUATIONS COMMON CORE ALGEBRA II Eercise #1: Solve each of the sstems of equations b elimination. (a) 3 9 (b) You should be ver familiar with solving two-b-two sstems of linear equations (two equations and two unknowns). These linear sstems serve as the basis for a field of math known as Linear Algebra. Eercise #: Consider the three-b-three sstem of linear equations shown below. Each equation is numbered in this first eercise to help keep track of our manipulations. (1) () (3) z z z 14 (a) The addition propert of equalit allows us to add two equations together to produce a third valid equation. Create a sstem b adding equations (1) and () and (1) and (3). Wh is this an effective strateg in this case? (b) Use this new two-b-two sstem to solve the three-b-three. Modified from emathinstruction, RED HOOK, NY 1571, 015

27 Just as with two b two sstems, sometimes three-b-three sstems need to be manipulated b the multiplication propert of equalit before we can eliminate an variables. Eercise #3: Consider the sstem of equations shown below. Answer the following questions based on the sstem. 4 3z 6 4 z 38 (b) Solve the two-b-two sstem from (a) and find the final solution to the three-b-three sstem. 5 7z 19 (a) Which variable will be easiest to eliminate? Wh? Use the multiplicative propert of equalit and elimination to reduce this sstem to a two-b-two sstem. 7 Modified from emathinstruction, RED HOOK, NY 1571, 015

28 You can easil check our solution to an sstem of equations b storing our answers for each of the variables and making sure the make each of the equations true. This is especiall useful if ou get a multiple choice question on this topic. Use the storing method to solve the problem below. 8 Eercise #4: Which ordered triple represents the solution to the sstem of equations? 3z 11 4z 3 5 z 18 a. (-1,1,-4) b. (1,3,-) c. (3,8,1) d.(,-3,0) Eercise #5: Solve the sstem of equations shown below. Show each step in our solution process. 4 3z 3 5 3z 37 4z 7 These are challenging problems onl because the are long. Be careful and ou will be able to solve each one of these more comple sstems. Modified from emathinstruction, RED HOOK, NY 1571, 015

29 9 LESSON #47 - SYSTEMS OF LINEAR EQUATIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Solve the following sstems of equations algebraicall. (a) (b) Show that 10, 4, and z 7 is a solution to the sstem below without solving the sstem formall. z 5 4 5z 1 8z 3 3. Solve the following sstem of equations. Carefull show how ou arrived at our answers. 4 z 1 z z 70 Modified from emathinstruction, RED HOOK, NY 1571, 015

30 4. Algebraicall solve the following sstem of equations. There are two variables that can be readil eliminated, but our answers will be the same no matter which ou eliminate first. 5 z z 31 3 z Algebraicall solve the following sstem of equations. This sstem will take more manipulation because there are no variables with coefficients equal to 1. 3 z z z 50 Modified from emathinstruction, RED HOOK, NY 1571, 015

31 31 LESSON #48 FOCUS AND DIRECTRIX OF A PARABOLA PART 1 COMMON CORE ALGEBRA II Materials: 1 sheet patt paper, 1 pencil, 4 colored pencils, ruler 1. Use ruler to draw a straight line with one to two ruler-widths from the bottom of the patt paper. Fold paper in half, folding line upon itself making a crease, mark a point above the line on this crease 3. Make several creases in which the line coincides with the point 4. Using a one of the colored pencils outline the shape that the creases form 5. Draw a sketch of the line, the point, and the shape formed in the tetbo below. DEFINITION OF A PARABOLA A parabola is a special curve shaped like an arch. An point on a parabola is at an equal distance from.. * a fied straight line,, and *a fied point, the Label each of the above in the tetbo with a second color. The ais of smmetr is the line that divides a parabola into two parts that are mirror images. The verte of a parabola is the point at which the parabola intersects the ais of smmetr. On the coordinate plane, the coordinates of the verte are represented b the general point (h,k). The verte is also the directl between the focus and directri. Draw the ais of smmetr and the verte in the tetbo above with a third color and label them. The directed distance from the verte to the focus is represented b the variable, p. This new variable p is one ou'll need to be able to work with when writing equations of parabolas. Label p in the tetbo with the fourth color. Modified from emathinstruction, RED HOOK, NY 1571, 015

32 3 GEOMETRIC DEFINITION OF A PARABOLA A parabola is the set of all points (,) in a plane that are equidistant from a fied line (directri) and a fied point (focus) not on the line. The standard form of the equation of a parabola with a vertical ais of smmetr when the verte (h,k) and the p value are known is If the lead coefficient is positive: the parabola opens up. If the lead coefficient is negative: the parabola opens down. 1 ( ) h k 4 p When the ais of smmetr is a horizontal ais, the formula is: 1 ( ) k h 4 p If the lead coefficient is positive: the parabola opens right. If the lead coefficient is negative: the parabola will open left. The important difference in the two equations is which variable is squared: for regular (vertical) parabolas, is squared; for sidewas (horizontal) parabolas, is squared. Note: the values of h and k are switched as well so that the -value of the verte is with the -variable. For the first da, we will be focusing on parabolas with vertical directries Eercise #1: For each of the following equations of a parabola (a) state whether it opens up or down and identif the (b) verte, (c) p-value, (d) focus, and (e) the equation of the directri ( 3) 4 1 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Modified from emathinstruction, RED HOOK, NY 1571, 015

33 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri 3. 8( 8) ( 7) (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Modified from emathinstruction, RED HOOK, NY 1571, 015

34 34 4. ( 3) (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Eercise #: Write the equation of a parabola with each of the following qualities. It ma be helpful to draw a sketch of the parabola first. a) Write the equation of a parabola whose verte is (,0) and directri is = 6. b) Write the equation of a parabola whose directri is = and focus is (-1,8). Modified from emathinstruction, RED HOOK, NY 1571, 015

35 35 LESSON #48 FOCUS AND DIRECTRIX OF A PARABOLA PART 1 COMMON CORE ALGEBRA II HOMEWORK For each of the following equations of a parabola (a) state whether it opens up or down and identif the (b) verte, (c) p-value, (d) focus, and (e) the equation of the directri ( 3) 1 8 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri 1. 4 ( 4) 0 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri 3. 1( 6) ( 4) (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Modified from emathinstruction, RED HOOK, NY 1571, 015

36 36 4. ( 1) 4 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri For the following problems, it ma be helpful to sketch the parabola on graph paper. 5. Write the equation of a vertical parabola whose verte is (-1,4) and p-value is Write the equation of a parabola whose verte is (3,9) and focus is (3,6). 7. Write the equation of a parabola whose verte is (0,1) and directri is = Write the equation of a parabola whose directri is = -1 and focus is (,-5). Modified from emathinstruction, RED HOOK, NY 1571, 015

37 37 LESSON #49 FOCUS AND DIRECTRIX OF A PARABOLA PART COMMON CORE ALGEBRA II Eercise #1: Write the general equation of a parabola that has a vertical ais of smmetr. Eercise #: Write the general equation of a parabola that has a horizontal ais of smmetr. Eercise #3: For each of the following equations of a parabola (a) state whether it opens right, left, up, or down and identif the (b) verte, (c) p-value, (d) focus, and (e) the equation of the directri ( ) 3 4 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri. 1 5 ( 3) 16 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri Modified from emathinstruction, RED HOOK, NY 1571, 015

38 ( 6) (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 4. ( 1) 8 4 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 5. Write an equation for the set of points which are equidistant from the origin and the line = Write the equation of a parabola whose focus is (3,1) and directri is = Write an equation for the set of points which are equidistant from (4,-) and the line = 4. Modified from emathinstruction, RED HOOK, NY 1571, 015

39 LESSON #49 FOCUS AND DIRECTRIX OF A PARABOLA PART COMMON CORE ALGEBRA II HOMEWORK For each of the following equations of a parabola (a) state whether it opens right, left, up, or down and identif the (b) verte, (c) p-value, (d) focus, and (e) the equation of the directri ( 3) 1 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri ( 3) 0 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 10. ( 1) 1 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri Modified from emathinstruction, RED HOOK, NY 1571, 015

40 ( 3) 8( 5) (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 1. Write the equation of the parabola with focus (1,6) and directri = Write an equation for the set of points which are equidistant from (0,) and the -ais. 14. Write the equation of the parabola with focus (-,0) and directri the -ais. 15. Write the equation for the set of points equidistant from (,3) and = -3 Modified from emathinstruction, RED HOOK, NY 1571, 015

### LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

### LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

### LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important

### LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

### 8.1 Exponents and Roots

Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents

### Lesson 9.1 Using the Distance Formula

Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

### 17. f(x) = x 2 + 5x f(x) = x 2 + x f(x) = x 2 + 3x f(x) = x 2 + 3x f(x) = x 2 16x f(x) = x 2 + 4x 96

Section.3 Zeros of the Quadratic 473.3 Eercises In Eercises 1-8, factor the given quadratic polnomial. 1. 2 + 9 + 14 2. 2 + 6 + 3. 2 + + 9 4. 2 + 4 21. 2 4 6. 2 + 7 8 7. 2 7 + 12 8. 2 + 24 In Eercises

### UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS

Answer Key Name: Date: UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS Part I Questions. Which of the following is the value of 6? () 6 () 4 () (4). The epression is equivalent to 6 6 6 6 () () 6

### Keira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:

Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic

### 1.2 Functions and Their Properties PreCalculus

1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

### Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.

NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations

### Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1

Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab

### 6.4 graphs OF logarithmic FUnCTIOnS

SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS

### 9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson

Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric

### Vertex. March 23, Ch 9 Guided Notes.notebook

March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function

### = x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

### Unit 10 - Graphing Quadratic Functions

Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

### LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II

1 LESSON #1 - BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarif concepts and remove ambiguit from the analsis of problems. To achieve

### Algebra 2 CPA Summer Assignment 2018

Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

### Using Intercept Form

8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of

### 1. Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. (c) ( )

Name: Date: QUADRATIC FUNCTION REVIEW FLUENCY Algebra II 1. Without the use of our calculator, evaluate each of the following quadratic functions for the specified input values. (a) g( x) g g ( 5) ( 3)

### ALGEBRA 2 NY STATE COMMON CORE

ALGEBRA NY STATE COMMON CORE Kingston High School 017-018 emathinstruction, RED HOOK, NY 1571, 015 Table of Contents U n i t 1 - Foundations of Algebra... 1 U n i t - Linear Functions, Equations, and their

### Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.

Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have

### Power Functions. A polynomial expression is an expression of the form a n. x n 2... a 3. ,..., a n. , a 1. A polynomial function has the form f(x) a n

1.1 Power Functions A rock that is tossed into the water of a calm lake creates ripples that move outward in a circular pattern. The area, A, spanned b the ripples can be modelled b the function A(r) πr,

### SECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z

8-7 De Moivre s Theorem 635 B eactl; compute the modulus and argument for part C to two decimal places. 9. (A) 3 i (B) 1 i (C) 5 6i 10. (A) 1 i 3 (B) 3i (C) 7 4i 11. (A) i 3 (B) 3 i (C) 8 5i 12. (A) 3

### Maintaining Mathematical Proficiency

Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.

### Lesson #33 Solving Incomplete Quadratics

Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

### Laurie s Notes. Overview of Section 3.5

Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its

Chapter 11 Quadratic Functions Mathematical Overview The relationship among parabolas, quadratic functions, and quadratic equations is investigated through activities that eplore both the geometric and

### 3 Polynomial and Rational Functions

3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental

### KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

### 1.2 Functions and Their Properties PreCalculus

1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

### Maintaining Mathematical Proficiency

Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +

### Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

### A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.

1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function

### Reteaching (continued)

Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression

### Polynomial and Rational Functions

Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;

### MATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)

MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the

### Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

### CHAPTER 2 Polynomial and Rational Functions

CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............

### Fair Game Review. Chapter 9. Find the square root(s) ± Find the side length of the square. 7. Simplify Simplify 63.

Name Date Chapter 9 Find the square root(s). Fair Game Review... 9. ±. Find the side length of the square.. s. s s Area = 9 ft s Area = 0. m 7. Simplif 0. 8. Simplif. 9. Simplif 08. 0. Simplif 88. Copright

### 1.7 Inverse Functions

71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

### 4.2 Parabolas. Explore Deriving the Standard-Form Equation. Houghton Mifflin Harcourt Publishing Company. (x - p) 2 + y 2 = (x + p) 2

COMMON CORE. d Locker d LESSON Parabolas Common Core Math Standards The student is epected to: COMMON CORE A-CED.A. Create equations in two or more variables to represent relationships between quantities;

### INTEGER EXPONENTS HOMEWORK. 1. For each of the following, determine the integer value of n that satisfies the equation. The first is done for you.

Name: Date: INTEGER EXPONENTS HOMEWORK Algebra II INTEGER EXPONENTS FLUENCY. For each of the following, determine the integer value of n that satisfies the equation. The first is done for ou. n = 8 n =

### Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

### Algebra 2 Honors Summer Packet 2018

Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

### 4 The Cartesian Coordinate System- Pictures of Equations

The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean

### f(x) = 2x 2 + 2x - 4

4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms

### Section 2.5: Graphs of Functions

Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree

### Instructor: Imelda Valencia Course: A3 Honors Pre Calculus

Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression

### Ready To Go On? Skills Intervention 6-1 Polynomials

6A Read To Go On? Skills Intervention 6- Polnomials Find these vocabular words in Lesson 6- and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading

### Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.

Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of

### UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

### 2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.

9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the -intercepts of a graph in Section., it follows that the -intercept

### STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

### Polynomial and Rational Functions

Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

### 5. Perform the indicated operation and simplify each of the following expressions:

Precalculus Worksheet.5 1. What is - 1? Just because we refer to solutions as imaginar does not mean that the solutions are meaningless. Fields such as quantum mechanics and electromagnetism depend on

### y = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is

Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,

### Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.

Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to

### 3.7 InveRSe FUnCTIOnS

CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

### Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general

A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate

### Inverse of a Function

. Inverse o a Function Essential Question How can ou sketch the graph o the inverse o a unction? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to

### Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and

### Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

### MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand

### Exponential and Logarithmic Functions

Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs

### Name Class Date. Solving by Graphing and Algebraically

Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem

### Common Core State Standards for Activity 14. Lesson Postal Service Lesson 14-1 Polynomials PLAN TEACH

Postal Service Lesson 1-1 Polynomials Learning Targets: Write a third-degree equation that represents a real-world situation. Graph a portion of this equation and evaluate the meaning of a relative maimum.

### 2.1 The Rectangular Coordinate System

. The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table

### Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

### Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane?

10.7 Circles in the Coordinate Plane Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? The Equation of a Circle with Center at the Origin Work

### Cubic and quartic functions

3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving

### Module 3, Section 4 Analytic Geometry II

Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related

### c) domain {x R, x 3}, range {y R}

Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..

### + = + + = x = + = + = 36x

Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

### UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x

5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able

### Math 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions

1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains

### Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

### Equations for Some Hyperbolas

Lesson 1-6 Lesson 1-6 BIG IDEA From the geometric defi nition of a hperbola, an equation for an hperbola smmetric to the - and -aes can be found. The edges of the silhouettes of each of the towers pictured

### 1.2. Characteristics of Polynomial Functions. What are the key features of the graphs of polynomial functions?

1.2 Characteristics of Polnomial Functions In Section 1.1, ou explored the features of power functions, which are single-term polnomial functions. Man polnomial functions that arise from real-world applications

### CALCULUS BASIC SUMMER REVIEW

NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=

### Course 15 Numbers and Their Properties

Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

### Methods for Advanced Mathematics (C3) Coursework Numerical Methods

Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on

### 4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the

Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

### Algebra 1 Unit 9 Quadratic Equations

Algebra 1 Unit 9 Quadratic Equations Part 1 Name: Period: Date Name of Lesson Notes Tuesda 4/4 Wednesda 4/5 Thursda 4/6 Frida 4/7 Monda 4/10 Tuesda 4/11 Wednesda 4/12 Thursda 4/13 Frida 4/14 Da 1- Quadratic

### Modeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models?

.7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match

Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker

### Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents

Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE

### REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES

Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.

### RELATIONS AND FUNCTIONS through

RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or

### 13. x 2 = x 2 = x 2 = x 2 = x 3 = x 3 = x 4 = x 4 = x 5 = x 5 =

Section 8. Eponents and Roots 76 8. Eercises In Eercises -, compute the eact value... 4. (/) 4. (/). 6 6. 4 7. (/) 8. (/) 9. 7 0. (/) 4. (/6). In Eercises -4, perform each of the following tasks for the

### 2 nd Semester Final Exam Review Block Date

Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict

### 12x y (4) 2x y (4) 5x y is the same as

Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents