The x- Intercepts (The Zeroes) The values of the function when y = 0 (, 0) & (3,0) The Role of the Discriminant: The Nature and Number of Roots root b ac = 0 roots b ac > 0 0 roots b ac < 0 (i) (ii) (iii) (iv) (v) Solving a Quadratic Function means finding the x-intercepts. Graph Factor to solve for the Zeroes a. Sum Product b. Decomposition c. Difference of Squares Complete the Square Use the Quadratic Formula b ± b ac a Use the TI-83 to find the zeroes (, -) Transformations of ± a y = x ( ) ( ) y k = x h VT is k HT is h VS is ± a A of S is h Vertex is ( h, k) The Vertex (The Turning Point) The x-coordinate gives you the axis of symmetry & the horizontal translation. The x-coordinate gives you the value of the minimum or maximum & the vertical translation. To Find the Vertex Use the Vertex Formula b a And then solve for y. The Pattern for the Curve once you know the vertex. From the vertex: Over up Over, up Over 3, up 9 When there is a vertical stretch: From the vertex: Over up vertical stretch Over, up vertical stretch Maximum or Minimum? Three Forms of the Quadratic Function General Form y = ax + bx + c If a is Negative, the parabola opens downwards. (Maximum) If a is positive, the parabola opens upwards. (Minimum) Happy Parabola a > 0 Sad Parabola a < 0 The Mapping Rule ( x + h ± ay k ) ( x, y), + Standard Form y = ± a x h + ( ) k Transformational Form ± a ( ) ( ) y k = x h
Review Unit Test : Quadratics Part Selected Response, Circle the correct answer. The x-intercepts for the equation, x + 6x 8 = 0, are, and 9 b) -3 and 6 c) and -9 b) -3 and 6 d) and 9 The equation y = ( x ) + in general form is: ( y = ) = ( x ) b) ( y + ) = ( x ) c) y = x + 6x d) y = x + 8x 3 A quadratic function f ( x) has a vertex at (-,0). What is the nature of the roots of the equation ( x) f =0? two equal imaginary roots b) two equal real roots c) two unequal imaginary roots d) two unequal real roots. Which of the following is a perfect square? x + 5x + 0 b) x x + 9 c) x + 6x 9 d) x + 5x + 0 5 The y-intercept for the quadratic equation, y = 3x 6x + is: b) - c) d) 6 What is the equation of the axis of symmetry of the parabola represented by the function with roots of -6 and 0? 8 b) c) 0 d) 7 Simplify 75 5 3 b) 5 3 c) ± 5 3 d) 5 3 8 If the quadratic equation y = ax + bx + c has no x-intercepts, which of the following is true? x b = ac b) b ac > 0 c) b ac = 0 d) b ac < 0 9 Simplify 0 60 b) i 60 c) ± 5 3 d) i 5 0 What is the transformational form of the function y = ( x + 3)? y = x x 3 b) ( y + ) = ( x + 3) c) ( y ) = ( x + 3) d) ( y + ) = ( x + 3)
Part Extended Response Which term must be added to x 5x to complete the square? What is the initial height of the ball thrown for this parabolic path? y = x + 5x + 3 3 The quadratic equation below was solved using the quadratic formula. Which step contains the first mathematical error and explain why? Step : ± ( ) () ()( 3) Step : ± 6 Step 3: ± Step : 3, Convert each of the following functions into a different form. (Each question must represent a different form. ( y ) = ( x + ) b) y = 3( x + ) 5 Determine the number and nature of the roots for each function: y = x 35x + 8 b) y = x + x + c) y = 6x x 6 Determine the nature of the roots of the following functions. y = x + 8x + b) 8 y = x + 8x + c) y = x + x + 50 7 Change the following functions into transformational form. y = x + x + 6 b) y = x + x + c) y = x 6 x 5 8 Change the following functions into general form: ( y 3) = ( x ) b) y = ( x + )
Part 3: Additional Items Name TRANSFORMATIONAL FORM a y q = x p = x + 3 STANDARD FORM Formula ± ( ) ( ) y = ± a( x p) + q EXAMPLE ( y ) ( ) = ( x ) Vertex Transformations Axis of Symmetry HT VS VT HT VS VT GENERAL FORM y = ax + bx + c y y = 3x + x + Mapping Rule ( x, y) ( ) ( x, y) ( ) ( x, y) ( ) Maximum or Minimum y-intercept HT VS VT Roots
. For ( y 8) = ( x ), determine the following and then graph the function. VS VT HT R x b) Vertex c) Mapping Rule 0 5 y x d) The Range 6 6 e) The Axis of Symmetry. Write the equation of the following parabolas. 0 y b) A quadratic function passes through the point (5, -) and has a vertex of (0,). 5 x 6 6 3. Solve the following word problems. (Show all steps.) A stone thrown from the ground follows a path defined by h ( t) = t + t + 30 where t is the time in seconds and h is the height in metres above the ground. What is the maximum height that the stone reaches and at what time does it reach its maximum height.
. Factor each of the following polynomials. x + 7x + b) 5x 7x + 0 c) 36x 8 d) x x + 9 e) 3 8x y 35x y xy f) x x 5. Solve for x, in order to find the roots, by completing the square. x + 6x = 0 b) 5x 9x + = 0 Solutions:, 8,, yes b) (. 8) c) ( x, y) x +, y + 8 d) yε { y 8 } e) ( 6) ( 3) 5 y = x + b) ( y ) = x 3 3 sec., 8 m ( x + 3)( x + ) b) ( + 3)( x 5) c) 9(x + 3)(x 3) d) ( x 3) e) 7xy(x y 5x 3y) f) ( x ( x 3) 5 3 ± 0 b), 5