( 3, 5) and zeros of 2 and 8.

Transcription

1 FOM 11 T26 QUADRATIC FUNCTIONS IN VERTEX FORM DETERMINING QUADRATIC FUNCTIONS IN VERTEX FORM I) THE VERTEX FORM OF A QUADRATIC FUNCTION (PARABOLA) IS. To write a quadratic function in vertex form you must know the values of a, p and q. To determine these values you must know the coordinates of the vertex and one other point. The vertex has the coordinates V(p, q), therefore it gives you the values of p and q found within the vertex form. The other point has the coordinates P(x, y), therefore it gives you the values of x and y found within the vertex form. The other point could be an x-intercept which has the coordinates P(x, 0), or it could be a y-intercept which has the coordinates P(0, y). To write a quadratic function, determine the values of p and q from the vertex and the values of x and y from a point on the parabola. Substitute the values of p, q, x and y into then solve for the value of a. Then substitute the values of a, p and q are into the vertex form: to give the quadratic function. A) SAMPLE PROBLEMS : Study this example carefully. Be sure you understand and memorize the process used to complete it. 1) Write a quadratic function for a parabola that has a vertex of (, 4) and passes through the point ( 2, 1). 1: Determine the values of p and q from the vertex and x and y from a point on the parabola. ( ) p = and q = 4 P ( 2, 1) x = 2 and y = 1 V, 4 2: Substitute the values of p, q, x and y into then solve for a. 1 = a ( 2 ) = a ( 3) = a ( 9) = a ( 9) = a 9 9 a = 1 3 3: Substitute the values of a, p and q into to give the answer. a = 1 3 ; p = ; q = 4 y = 3 1 ( x ) y = 1 ( 3 x ) 2 4 ( 3, ) and zeros of 2 and 8. 2) Write the equation of the parabola that has a vertex at 1: Determine the values of p and q from the vertex and x and y from the point. V ( 3, ) p = 3 and q = There are two zeros given which can be written as x-intercepts having these coordinates: ( 2, 0) and ( 8, 0). Only one is required and either can be used. Using x- ( ) x = 2 and y = 0 intercept = 2, which has a y-coordinate is 0 making its coordinates: P 2, 0 Continued on the next page.

2 FOM 11 T26 QUADRATIC FUNCTIONS IN VERTEX FORM : Substitute the values of p, q, x and y into then solve for a. 0 = a ( 2 3) = a ( 2 + 3) = a ( ) 2 + = a ( 2) 2 = a 2 2 a = 1 3: Substitute the values of a, p and q into to give the answer. a = 1 ; p = 3; q = y = 1( x 3) 2 + y = 1 ( x + 3) 2 + B) REQUIRED PRACTICE 1: Page 417, 419 & 420: Questions 4, 11 & 12. SHOW THE PROCESS!! {Ans. Page 76} II) The process described above can be used to create a quadratic function that describes a real life structure such the path a soccer ball follows when kicked. In order to solve problems where you are required to write the quadratic function that defines a real life situation you must have the coordinates of the vertex and one other point. The SAMPLE PROBLEM given below outlines the steps required to create the quadratic function necessary to answer the questions in the problem. It is very important to remember that parabolas are symmetrical around the axis of symmetry, and that the axis of symmetry is described by the equation x = x-coordinate of the vertex. A) SAMPLE PROBLEM 1: Study these examples carefully. Be sure you understand and memorize the process used to complete it. 1) Solve the problem given in Example 4 on page 413. a) Model the shape of the cables with a quadratic function. 1: Sketch a diagram of the shape described in the problem. This will assist you in recognizing parabola and identifying the vertex and the point required create its quadratic function. 3: Determine the values of p and q from the vertex and x and y from the point. ( ) p = 2 and q = 20 P ( 4, 0) x = 4 and y = 0 V 2, 20 Continued on the next page.

3 FOM 11 T26 QUADRATIC FUNCTIONS IN VERTEX FORM : Substitute the values of p, q, x and y into then solve for a. 0 = a ( 4 2) = a ( 2) 2 20 = a ( 4) 20 4 = a 4 4 = a : Substitute the values of a, p and q into to give the answer. a = ; p = 2 ; q = 20 y = ( x 2) y = ( x 2) is the quadratic function that models the path of the ball. b) Determine any restrictions that must be placed on the domain and range of the function. Because negative time does not exist, the domain is { x 0 x 4,x R} { } Because the ball remains on or above the surface of the ground the range is y 0 y 20,y R c) What was the height of the ball at 1 second? 1: Determine the x-coordinate. The x-coordinate of y = ( x 2) represents time, thus x = 1: 2: Substitute the value of x into to give the answer. y = ( x 2) y = ( 1 2) y = ( 1) y = ( 1) + 20 y = + 20 y = 1 The ball is 1 m high at 1 second. B) REQUIRED PRACTICE 2: Page 420: Question 14, 1 & 17. SHOW THE PROCESS!! {Ans. Page 76}

4 FOM 11 T26 QUADRATIC FUNCTIONS IN VERTEX FORM DO NOT USE THESE GRIDS FOR YOUR ASSIGNMENT

5 FOM 11 T26 QUADRATIC FUNCTIONS IN VERTEX FORM - 2 ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER LAST then FIRST Name T26 QUADRATIC FUNCTIONS IN VERTEX FORM - 2 Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.) Answer these questions. REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! 1) Write the quadratic function of the parabola having a: (3. each) a) vertex of V 3, 1 b) vertex of V 4, 6 c) vertex of V 3, 3 ( ) and passes through the point (2, 3). ( ) and a y-intercept of 10. ( ) and has x-intercepts of 2 and 4. 2) Write the equation of this parabola. (.) 3) Complete the Your Turn questions a & c on page 41 of your Text. (11) Factor these expressions. If there are no factors state so. 4) 4x 2 2x + 3 (1) ) 2x (3) /31