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= m y-intercept= b General Form of Linear Equation are not zero. A By C such that A and B both What is a Function? A function is a relation that assigns a single element of R to each element of D. A working definition of a function is that it is a devise that assigns an output to every allowable input. The inputs make up the domain of the function. The outputs make up the range. A Function must pass the vertical Line Test Vertical Line Test Identifying the Domain and Range: Remember, in the real number system you not divide by zero or find the even root of a negative number can Even and Odd Functions A function y = f() is an even function of if f(-) = f() for every in the function s domain. A function is an odd function of if f(-) = - f() for every in the function s domain. Page
Absolute Value Think of the absolute value function as a piecewise function. The Greatest Integer Function f ( ) or f ( ) int( ) f ( ) if 0 if <0 The greatest integer function represents the greatest integer less than or equal to...4 4 Composition of Functions The composition f g of the functions f and g is defined by ( f g)( ) f ( g( )) The domain of domain of f. ( f g)( ) f ( g( )) consists of those s for which g() is in the Geometric Transformations: Shifts, Reflections, Stretches, and Shrinks Graph Shifting Formulas Vertical shifts of the graph of y f () y f ( ) c Shifts graph of y f () down c units y f ( ) c Shifts graph of y f () up c units Horizontal shifts of the graph of y f ( c) Shifts graph of y f () right c units y f ( c) Shifts graph of y f () left c units How to stretch or shrink a graph To stretch the parabola each y-coordinate by c. y vertically by a factor of c (c>0), we must multiply If you stretch the graph by a factor of two the new equation will be: y How to reflect a graph To reflect the graph of y=f() across the y-ais, we multiply each y coordinate by -. Reflection Formulas: With respect to the y-ais y f ( ) With respect to the -ais y f () The Parabola y a b c A parabola that opens in the positive y direction if a>0 and in the negative y direction if a<0. b b b The ais of symmetry is: The verte is at: (, f ( )) a a a Page
POLYNOMIALS Polynomial Epression: n n n an an an... a a0 Polynomial Function: n n n f ( ) an an an... a a0 Polynomial Equation: n n n an an an... a a0 0 Rational Zeros Theorem Suppose all the coefficients in the polynomial function n n n f ( ) an an an... a a0 are integers. c If s a rational zero of f, where c and d have no common factors, then c is a factor d of a 0, and d is a factor of the leading coefficient a n. How to Solve f()= 0 using calculator or your own brain!!!!. Find the eact solution algebraically (often by factoring). Draw a complete graph a) Use ZOOM-IN b) Use SOLVE Steps for Solving a Problem. Find an algebraic representation involving variables.. Draw a complete graph of the function. Find the domain and range 4. Determine the values that make sense in the given situation 5. Draw a graph of the problem situation 6. Solve the problem using appropriate methods For instance: Solve 4 0 Factors of c:,,, 4, 6, Factors of d:, c Possible zeros:,,, 4, 6,,, d Look at the graph to see the zeroes must be between - and - or and 4. So f ( / ) 0 ( / ) ( ) So ( ) is a factor. By division ( )( 4) 0 Use the Quadratic Formula to find 5 Equations for Circles in the Plane Circle is the set of points in a plane whose distance from a fied point in the plane is a constant. The fied point is the center of the circle. The constant distance is the radius of the circle. Equation: ( h) ( y k) r Page
Inverse Relations and Functions: Inverse Relation: Let R be a relation. The inverse relation R of R consists of all those ordered pairs (b,a) for which (a,b) belongs to R. So the domain of R = the range of R and the range of R = the domain of R. Horizontal Line Test : The inverse relation R of the relation R is a function if and only if every horizontal line intersects the graph of R in at most one point. Notice that the inverse of f ( ) 6 is not a function since f() fails the horizontal line test. One-to-One: The inverse f of a function f is a function if and only if f is a one-to-one function. Eponential Functions: Definition: Let a be a positive real number other than. The function f ( ) a whose domain is (, ) and whose range is ( 0, ) is the eponential function with base a. Trigonometric Functions: Unit Circle Page 4
Graph of the Sine Curve: y = sin() Graph of the Cosine Curve: y = cos() Graph of the Tangent Curve: Conic Sections y = tan() Circle Ellipse (h) Parabola (h) Hyperbola (h) Definition: A conic section is the intersection of a plane and a cone. Ellipse (v) Parabola (v) Hyperbola (v) By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or Page 5
hyperbola; or in the special case when the plane touches the verte: a point, line or intersecting lines. Point Line Double Line The General Equation for a Conic Section: A + By + Cy + D + Ey + F = 0 Assignment # Calculus Simplify the following algebraic and numeric epressions.... 7 (9 7 5) = 5 7 = (4 5) = 8. (4i ) (i ) = 9. (4i)(i ) = 0. 5[4( y ) ( y )] =. = 4. 0 0 (5 ) a = 5. (7 y) ( 5 y) = 6. (7 y)( 5 y) = 7. (4i ) (i ) = Simplify without a calculator, giving answer in eact form (not decimal). In your answer, epress all eponents as positive values and convert any fractional powers to radical form... 4. y z = z y t t 5 5t 8 5t Page 6
5. 6. 8 6 8 y z ( ) 6 y z 7. 8. 9. ( ) (6 ) 4 0 4 ( ) 7 98 Find the eact value without your calculator (no decimal answers). 0. 700 7. ( 6) ( 5). 4 5. 0 7 4. 6 5 Solve each equation algebraically; verify your solution by substituting in the original equation. 5. ( 7) 5 8. 4 5 6. y 4 8 4 4. 5 7. 8. 9. 0. 5 7 0 ( ) ( 7)( ) 5 5. solve for in terms of. 6. c, where a b a 0, b 0, b a, solve for.. Solve by completing the square: 40 0. 5 a 7. r s, solve for r. Find the solution to the system of equations. y 7 8. 56y 9 9. y 98y 4 Page 7
4. Determine the slope between the points (4, -) and (-6, 4). 40. Determine the slope of the line -y = -. 4. Write in slope-intercept form the equation of the line containing the point (-, ) and parallel to the given line y = + 4. 4. Write in slope-intercept form the equation of the line containing the point (4, 5) and perpendicular to the given line y = 6. You should know how to quickly sketch the graphs of these five basic parent functions: a) y b) y c) y d) y e) y 4. From the parent graph of y ( ) 5 and graph the function. y describe the shift to obtain the new graph of 44. From the parent graph of y describe the shift to obtain the new graph of y and graph the function. 45. State whether the given set of points is a relation or a function{( }. 46. 47. State the domain and range of the function h ( ) and vertical and horizontal asymptotes if any eist. 48. Find the domain, range, zero(s), and y-intercept of f ( ) and verify by graphing. 49. Find the domain, range, zero(s), and y-intercept of g( ) 4 and verify by graphing. 50. Given 5. Given f ( ) find its inverse f ( ) and then sketch the graph of both. g( ) 4 find its inverse g ( ) and then sketch the graph of both. For #5 and 5 use the following: 5. Find f ( g( )) f g 4 ( ) ; ( ) 59. 5. Find g( f ( )) Factor the following: 58. 9y 400 9 8 60. 6. 4y 4z 6 7 Page 8
6. 5 0 y y Page 9
Summer Reading AP Calculus: 00 Complete the indicated operation to simplify the polynomials. Rational answers should have a common denominator. 6 8 5 4 6. 4 5 0 64. 8 64 9 4 65. 5 6 65. 6 66. What are the lengths of the missing sides of a 0 60 90 triangle if the longer leg of the triangle is 8 centimeters? 66. The hypotenuse of a 0 60 90 triangle is inches. Find the measures of the other two sides. 68. What is the length of the hypotenuse of a 45 45 90 triangle if one leg measures 9 centimeters? 69. The leg of a 45 45 90 triangle is 4 centimeters. Find the measures of the other two sides. 70. If the radius of a circle is 6 centimeters, what is its eact circumference? 7. If the radius of a circle is 6 centimeters, what is its area? 7. What is the area of a triangle with base of 7 cm and altitude to the base of 4 cm? 7. If the base of a parallelogram is 5 inches and altitude to the base is one third of the base, what is the area of this parallelogram? 0