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Functions, Quadratic Equations, and Inequalities - Analytic Of Mathematics Grade X Semester 1 Chapter 2

This educational content covers the understanding of functions, quadratic equations, and inequalities, including plotting graphs and solving quadratic equations. Designed for Grade X students.

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Functions, Quadratic Equations, and Inequalities - Analytic Of Mathematics Grade X Semester 1 Chapter 2

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  1. FUNCTIONS, QUADRATIC EQUATIONS, AND INEQUALITIES RatihKusumawati (4101408037) RatriRahayu (4101408041) HimmatulUlya (4101408042) Designed by HimmatulUlya 085641215539 Analytic Of MathematicS curiculum Grade X SEMESTER 1

  2. Chapter 2 Functions Quadratic Equations and Inequalities Plotting Graphs of Quadratic Equations Exit Designed by HimmatulUlya 085641215539

  3. FUNCTIONS EXERCISE HOME Designed by HimmatulUlya 085641215539

  4. Functions Understanding of functions: Suppose we have two non-zero sets A and B; A function or mapping f from A to B is the pairing of every element in A to exactly one element in B. Designed by HimmatulUlya 085641215539

  5. In the mapping f from A to B, element is mapped (paired) by function f to an element in B, written f(a). f The mapping from A to B which maps every by function is denoted as follows. a b = f(a) A B Designed by HimmatulUlya 085641215539

  6. Domain (D) Set A, which is the original set of all mapping elements. Function Co-domain (C) Set B, the target set of the mapping. f Range (R) The set of all maps of set A. The range is a subset of co-domain B. a f(a) A B Designed by HimmatulUlya 085641215539

  7. Simple and Quadratic Algebraic Functions Constant Functions Identity Functions Modulus Functions Linear Functions Quadratic Functions Designed by HimmatulUlya 085641215539

  8. Constant Functions A constant function is defined as a function that maps every element in a domain to a single common value (a constant). Designed by HimmatulUlya 085641215539

  9. Example: Plot the graph of function . , Answer: The formula of this function is . The graph: Y 4 Based on the figure, the graph of the function f(x)=3 is parallel to the X axis and passes through coordinates (0,3). 3 2 1 X 0 1 2 3 4 Designed by HimmatulUlya 085641215539

  10. Identity Functions The identity function is a function that maps every element in a domain to itself. Designed by HimmatulUlya 085641215539

  11. Example: Plot the graph of function . , Answer: The formula of this function is . Take several points in , and so on. Y 2 1 X -1 0 1 2 -1 • The graph is the line of equation y=x Designed by HimmatulUlya 085641215539

  12. Modulus Functions A modulus function is a function that maps every element in a domain to positive value or zero. “ “ is a common symbol to denote an absolute value. Designed by HimmatulUlya 085641215539

  13. Example: Plot the graph of function . , Answer: Y 4 3 2 1 X -3 0 -2 -1 1 2 3 Designed by HimmatulUlya 085641215539

  14. Linear Functions A linear function is a function that maps every to a form , with , a and b are constants. The formula of this function is . The graph is the line of equation . Designed by HimmatulUlya 085641215539

  15. Example: , Plot the graph of function . Answer: Y 4 3 2 1 X 1 0 2 -2 -1 Designed by HimmatulUlya 085641215539

  16. Quadratic Functions A quadratic function is a function of a general form Quadratic functions have a special graph shape, namely parabola. Designed by HimmatulUlya 085641215539

  17. Example: Plot the graph of a function . , Answer: Y 4 (2,4) (-2,4) 3 2 1 (1,1) (-1,1) (0,0) X 2 0 -2 -1 1 Designed by HimmatulUlya 085641215539

  18. QUADRATIC EQUATION AND INEQUALITIES 2 3 1 Designed by RatriRahayu 085740831948

  19. Designed by RatriRahayu 085740831948

  20. QUADRATIC EQUATION AND THEIR SOLUTIONS Designed by RatriRahayu 085740831948

  21. The general form of quadratic equation is x is called a variable, a is called a coefficient of x2, b is called a coefficient of x, c is a constant (fixed term).

  22. Below are a few examples of quadratic equations with a as the coefficient of x2, b as the coefficient of x and c as the constant of ax2+bx+c=0 (1) x2+40x-21000=0 a=1, b=40 and c = -21000 (2) y-y2=0 a=-1, b=1 and c = -0 (3)(p+1)t2+2pt -2p+1=0 a=(p+1), b=2p and c = -2p+1 Designed by RatriRahayu 085740831948

  23. SOLVING QUADRATIC EQUATION Solving quadratic equation ax2+bx+c=0 means determining values of x that satisfy the quadratic equation. Values of x that satisfy the quadratic equation are called the roots or solution the quadratic equation. Designed by RatriRahayu 085740831948

  24. THE ROOTS OF QUADRATIC EQUATION The roots of quadratic equation can be determine by: FACTORIZATION COMPLETING THE SQUARE QUADRATIC FORMULA Designed by RatriRahayu 085740831948

  25. FACTORIZATION Solving quadratic equation with factorization, we use the following multiplication attribute. The null factor law If ab = 0, then a=0 or b=0 Designed by RatriRahayu 085740831948

  26. Case a=1 The general form of quadratic equations becomes x2+bx+c=0. We will transform this equation into the form (x+α)(x+β)=0. x2+bx+c=(x+α)(x+β) = x2+ α x+ βx + α β = x2+ (α+β)x + αβ Use the “null factor law” to solve the equation. (x+α)(x+β)=0 x+α=0 or x+β=0 x=-α or x=-β Designed by RatriRahayu 085740831948

  27. Determine the roots of the following equation by factorization - It’s in the correct form But can we FACTORISE ? Guide number 3 1x3 Subtract to 2 Yes! Perfect EXAMPLE Factorised Solved or Thus the solutions (roots) are x=-1 and x=3 Designed by RatriRahayu 085740831948

  28. Case a ≠ 1 With α+β=b and αβ=ac. Designed by RatriRahayu 085740831948

  29. - Subtract to 83 No Good Can we FACTORISE ? Guide number 84 1x84 2x42 3x28 4x21 6x14 7x12 Subtract to 40 No Good Subtract to 25 No Good YES we can Subtract to 17 Perfect! Group together Factorised or Solved Designed by RatriRahayu 085740831948

  30. COMPLETING THE SQUARE Solving the quadratic equations by factorization can yield solutions quickly if we can find the pair (α,β). But most often it is difficult in finding the pair, for example in the quadratic equation In order to do that, we can use another method which is slightly longer. This method is by completing the square. Designed by RatriRahayu 085740831948

  31. To solve by completing the square. If a ≠ 1, multiply both side of the equation by to get the equation in form of To complete the square on , take half cofficient of x and square it. Then add and subtract that number Take the roots and solve for x Designed by RatriRahayu 085740831948

  32. EXAMPLE Solve the following quadratic equation by completing the square. Multiply both sides of the equation by Solution: Add and subtract that number or The solution is . It is said that the equation has equal roots (twin roots) namely Designed by RatriRahayu 085740831948

  33. QUADRATIC FORMULA Solving the quadratic equation by completing the square always works. However, there is another way to solve the quadratic equation, namely by using quadratic formulas. The Quadratic Formula If a quadratic equation has a solution, they are given by Designed by RatriRahayu 085740831948

  34. EXAMPLE Solve the following quadratic equation by using the quadratic formula Solution: or Hence, the solution is or 2. Designed by RatriRahayu 085740831948

  35. Quadratic Solutions The number of real solutions is at most two. No solutions One solution Two solutions Designed by RatriRahayu 085740831948

  36. QUADRATIC INEQUALITIES AND THEIR SOLUTIONS Designed by RatriRahayu 085740831948

  37. The general form of quadratic inequalities is , , , , or , with , and EXAMPLES Is a quadratic inequalities with Is not a quadratic inequalities because it is equivalent to which is a linear inequality Designed by RatriRahayu 085740831948

  38. Example 1: Solve the quadratic inequality x2 – 5x + 6 > 0 graphically. Designed by RatriRahayu 085740831948

  39. Procedures: The corresponding quadratic function is y = x2 – 5x + 6 Step (1): Step (2): Factorize x2 – 5x + 6, we have y = (x – 2)(x – 3) ,i.e. y = 0, when x = 2 or x = 3. Step (3): Sketch the graph of y = x2 – 5x + 6. Step (4): Find the solution from the graph. Designed by RatriRahayu 085740831948

  40. y y = (x – 2)(x – 3) , y = 0, when x = 2 or x = 3.  x 0 2 3 Sketch the graph y =x2 – 5x + 6 . What is the solution of x2 – 5x + 6 > 0 ? Designed by RatriRahayu 085740831948

  41. so we choose the portion above the x-axis. 2 3 We need to solve x 2 – 5x + 6 > 0, y The portion of the graph above the x-axisrepresents y > 0 (i.e. x 2 – 5x + 6 > 0) x 0 The portion of the graph below the x-axis represents y < 0 (i.e. x 2 – 5x + 6 < 0) Designed by RatriRahayu 085740831948

  42. 2 3 y When x < 2, the curve is above the x-axis i.e., y > 0 x2 – 5x + 6 > 0 When x > 3, the curve is above the x-axis i.e., y > 0 x2 – 5x + 6 > 0 x 0 Designed by RatriRahayu 085740831948

  43. or From the sketch, we obtain the solution Designed by RatriRahayu 085740831948

  44. Graphical Solution: 0 2 3 Designed by RatriRahayu 085740831948

  45. SOLVING QUADRATIC INEQUALITIES BY USING THE TEST POINT METHOD There will be a time when we might forget the sequence of the sign each segment to the sides of the roots. The test its sign by substitusing its value into the left-hand side of the inequality. Suppose the result is positive, then the segment where the point is located is positive. Designed by RatriRahayu 085740831948

  46. EXAMPLE Determine the solution of the quadratic inequality SOLUTION: The solution of . They are not solutions of the inequality, but they divide the real number line in a natural way, pictured as follows. The products is positive or negative, for values orther than , depending on the signs of the factor . We tabulate signs in these intervals. + + - 3 Hence the solution of inequality is Designed by RatriRahayu 085740831948

  47. DISCRIMINANTS OF QUADRATIC EQUATIONS is The formula to solve the quadratic equation D is called the discriminant of quadratic equations. Discriminant can be used to distinguish various roots of quadratic equations. Designed by RatriRahayu 085740831948

  48. SUMS AND PRODUCTS OF ROOTS OF QUADRATIC EQUATIONS The roots of the quadratic equation are Designed by RatriRahayu 085740831948

  49. EXAMPLE • If and are roots of the quadratic equation , then • the equation whose roots are: is… SOLUTION: or Designed by RatriRahayu 085740831948

  50. Y X HOME PLOTTING GRAPHS OF QUADRATIC FUNCTION QUADRATIC FUNCTION GRAPH y = ax2 + bx + c EXERCISE Designed by Ratih K. 085726950408

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