What Does Your Quadratic Look Like? EXAMPLES
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1 What Does Your Quadratic Look Like? EXAMPLES 1. An equation such as y = x 2 4x + 1 descries a type of function known as a quadratic function. Review with students that a function is a relation in which each element of the domain is paired with exactly one element of the range. Remind them of the vertical line test for functions. 2. Definition of a Quadratic Function A quadratic function is a function that can e descried y an equation of the form y = ax 2 + x + c, where a Graphs of quadratic functions have certain common characteristics. For instance, they all have a general shape called a paraola. Note: You may wish to demonstrate using a three-dimensional model of a doule cone. Explain that a paraola is a conic section formed y the intersection of a doule cone and a plane that is parallel to the surface of the cone. 4. The tale and graph elow illustrate some other common characteristics of quadratic functions. x x 2 4x + 1 y -1 (-1) 2 4(-1) (0) (1) (2) (3) (4) (5) Notice the matching values in the y-column. See graph on next page.
2 y = x 2 4x + 1 X (2, -3) Axis of symmetry 2 5. Notice that in the y-column of the tale, -3 does not have a matching value. Also, -3 is the y-coordinate of the lowest point on the graph of y = x 2 4x + 1. For the graph of y = x 2 4x + 1, The lowest point, called the minimum point, has the coordinates (2, -3). The minimum or maximum point of a paraola is often called the vertex. 6. The vertical line containing the minimum point (2, -3), is called the axis of symmetry. The equation of the axis of symmetry for the graph aove is If the graph of any quadratic function is folded along the axis of symmetry, the two halves coincide. In other words, the two halves of the paraola are symmetric. (1, 4) 8. Example: Graph y = x 2 + 2x + 3 x -x 2 + 2x + 3 y -2 -(-2) 2 + 2(-2) (-1) 2 + 2(-2) (0) 2 + 2(-2) (1) 2 + 2(-2) (2) 2 + 2(-2) (3) 2 + 2(-2) (4) 2 + 2(-2) Axis of symmetry 1
3 9. The graph of y = -x 2 + 2x + 3 opens downward. The equation of the axis of symmetry is 1. The graph has a highest point, or maximum point, at (1, 4). 10. In general, a paraola will open upward and have a minimum point when the coefficients of y and x 2 have the same sign. It will open downward and have a maximum point when the coefficients of y and x 2 have the opposite sign. The maximum or minimum point of the graph always lies on the axis of symmetry. 11. Notice in the graph of y = -x 2 + 2x + 3, the axis of symmetry is halfway etween any two points have the same y-coordinate. Consider the points on the graph whose coordinates are (-1, 0) and (3, 0). From these coordinates, the equation of the axis of symmetry may e found as shown elow: 12. In general, the equation of the axis of symmetry for the graph of a quadratic function can e found y using the following rule Equation of Axis of Symmetry The equation of the axis of symmetry for y = ax 2 + x + c, where a 0, is. 13. Example: Find the equation of the axis of symmetry and the coordinates of the maximum or minimum point of the graph of y =x 2 x 6. Then use the information to draw the graph First, find the equation of the axis of symmetry. The equation of the axis of symmetry is Remind students that a is the coefficient of x 2 and is the coefficient of x. Next, find the coordinates of the maximum or minimum point. Since the coefficients of y and x 2 have the same sign, the graph of the function has a minimum point. The minimum point lies on the axis of symmetry. Since the axis of symmetry is 2 1, the
4 minimum point will have an x-coordinate of 2 1. Find the y-coordinate y sustituting 1 for x in y = x 2 x The coordinates of the minimum point are (, ). 2 4 y = x 2 x y = ( ) y = y = 4 Then, construct a tale. For the values of x choose some integers greater than 2 1, and some less than 2 1. This insures that points on oth sides of the axis of symmetry are plotted. Use this information to draw the graph. x x 2 x - 6 y -2 (-2) 2 (-2) (-1) 2 (-2) (0) 2 (-2) (1) 2 (-2) (2) 2 (-2) (3) 2 (-2) 6 0 Axis of symmetry 2 1 Minimum point 1 25 (, ) 2 4
5 14. Example: Find the coordinates of the maximum point for the graph of y = -2x 2 8x + 9. Since the coefficients of y and x 2 have different signs, the graph of the function as a maximum point. 8 2( 2) 2 First, find the equation of the axis of symmetry. Use a = 2 and = 8. The equation of the axis of symmetry is 2 Since the maximum point lies on the axis of symmetry, sustitute 2 for x in y = 2x 2 8x + 9. y = -2(-2) 2 8(-2) + 9 y = y = 17 The coordinates of the maximum point are (-2, 17).
6 Name: Date: Class: WHAT DOES YOUR QUADRATIC LOOK LIKE? WORKSHEET Find the equation of the axis of symmetry and the coordinates of the maximum or minimum point of the graph of each quadratic function. Students will write out a tale for each prolem and then draw the graph. 1. y = -x 2 + 5x y = x 2 + 2x 3. y = 3x 2 6x y = x 2 4x 5 5. y = x 2 x 6
7 WHAT DOES YOUR QUADRATIC LOOK LIKE? WORKSHEET KEY Find the equation of the axis of symmetry and the coordinates of the maximum or minimum point of the graph of each quadratic function. Then draw the graph. 1. y = -x 2 + 5x + 6 First, find the equation of the axis of symmetry. The equation of the axis of symmetry is Remind students that a is the coefficient of x 2 and is the coefficient of x. Next, find the coordinates of the maximum or minimum point. Since the coefficients of y and x 2 have different signs, the graph of the function has a maximum point. The maximum point lies on the axis of symmetry. Since the axis of symmetry is 2 5, the maximum point will have an x-coordinate of 2 5. Find the y-coordinate y sustituting 5 for x in y = -x 2 + 5x The coordinates of the minimum point are (, ). 2 4 y = -x 2 + 5x y = ( ) + 5( ) y = y = 4 Then, construct a tale. For the values of x choose some integers greater than 2 5, and some less than 2 5. This insures that points on oth sides of the axis of symmetry are plotted. Use this information to draw the graph.
8 x -x 2 + 5x + 6 y -2 -(-2) 2 + 5(-2) (-1) 2 + 5(-1) (0) 2 + 5(0) (1) 2 + 5(1) (2) 2 + 5(2) (3) 2 + 5(3) (4) 2 + 5(4) (5) 2 + 5(5) (6) 2 + 5(6) y = x 2 + 2x First, find the equation of the axis of symmetry. The equation of the axis of symmetry is Remind students that a is the coefficient of x 2 and is the coefficient of x. Next, find the coordinates of the maximum or minimum point. Since the coefficients of y and x 2 have the same sign, the graph of the function has a minimum point. The minimum point lies on the axis of symmetry. Since the axis of symmetry is -1, the minimum point will have an x-coordinate of -1. Find the y-coordinate y sustituting 1 for x in y = x 2 + 2x. y = x 2 + 2x 2 Y = ( 1) + 2( 1) The coordinates of the minimum point are (-1, -1). y = -1 Then, construct a tale. For the values of x choose some integers greater than -1, and some less than -1. This insures that points on oth sides of the axis of symmetry are plotted. Use this information to draw the graph.
9 x x 2 + 2x y -2 (-2) 2 + 2(-2) 0-1 (-1) 2 + 2(-1) -1 0 (0) 2 + 2(0) 0 1 (1) 2 + 2(1) 3 3. y = 3x 2 6x + 5 First, find the equation of the axis of symmetry. The equation of the axis of symmetry is Remind students that a is the coefficient of x 2 and is the coefficient of x. Next, find the coordinates of the maximum or minimum point. Since the coefficients of y and x 2 have the same sign, the graph of the function has a minimum point. The minimum point lies on the axis of symmetry. Since the axis of symmetry is 1, the minimum point will have an x-coordinate of 1. Find the y-coordinate y sustituting 1 for x in y = 3x 2 6x + 5. y = 3x 2 6x + 5 The coordinates of the minimum point are (1, 2). y = 3(1) 2 6(1) + 5 y = 2 Then, construct a tale. For the values of x choose some integers greater than 1, and some less than 1. This insures that points on oth sides of the axis of symmetry are plotted. Use this information to draw the graph. x 3x 2 6x + 5 y -1 3(-1) 2-6(-1) (0) 2-6(-0) (1) 2-6(1) (2) 2-6(2) (3) 2-6(3)
10 4. y = x 2 4x 5 First, find the equation of the axis of symmetry. The equation of the axis of symmetry is Remind students that a is the coefficient of x 2 and is the coefficient of x. Next, find the coordinates of the maximum or minimum point. Since the coefficients of y and x 2 have the same sign, the graph of the function has a minimum point. The minimum point lies on the axis of symmetry. Since the axis of symmetry is 2, the minimum point will have an x-coordinate of 2. Find the y-coordinate y sustituting 2 for x in y = x 2 4x 5. y = y = x 2 4x 5 The coordinates of the minimum point are (2, 9). y = (2) 2 4(2) 5 y = 9 Then, construct a tale. For the values of x choose some integers greater than 2, and some less than 2. This insures that points on oth sides of the axis of symmetry are plotted. Use this information to draw the graph. x x 2 4x 5 y 0 (0) 2-4(0) (1) 2-4(1) (2) 2-4(2) (3) 2-4(3) (4) 2-4(4) 5-5
11 5. y = x 2 x 6 First, find the equation of the axis of symmetry. The equation of the axis of symmetry is Remind students that a is the coefficient of x 2 and is the coefficient of x. Next, find the coordinates of the maximum or minimum point. Since the coefficients of y and x 2 have the same sign, the graph of the function has a minimum point. The minimum point lies on the axis of symmetry. Since the axis of symmetry is 2 1, the minimum point will have an x-coordinate of 2 1. Find the y-coordinate y sustituting 2 1 for x in y = x 2 x The coordinates of the minimum point are (, 6 ). 2 4 y = x 2 x 6 y = ( 2 1 ) 2 4( 2 1 ) 5 Then, construct a tale. For the values of x choose some integers greater than 2 1, and some less than 2 1. This insures that points on oth sides of the axis of symmetry are plotted. Use this information to draw the graph. y = x x 2 x 6 y 0 (0) 2 - (0) (1) 2 - (1) (2) 2 - (2) (3) 2 - (3) (-2) 2 - (-2) 6 0
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13 Student Name: Date: WHAT DOES YOUR QUADRATIC LOOK LIKE CHECKLIST 1. On questions, (1 5), did the student find the axis of symmetry? a. All five (25 points). Four of the five (20 points) c. Three of the five (15 points) d. Two of the five (10 points) e. One of the five (5 points) 2. On questions, (1 5), did the student find the coordinates of the maximum or minimum point of the graph? a. All five (25 points). Four of the five (20 points) c. Three of the five (15 points) d. Two of the five (10 points) e. One of the five (5 points) 3. On questions, (1 5), did the student write out a tale for the graph? a. All five (25 points). Four of the five (20 points) c. Three of the five (15 points) d. Two of the five (10 points) e. One of the five (5 points) 4. On questions, (1 5), did the student draw the graph correctly? a. All five (25 points). Four of the five (20 points) c. Three of the five (15 points) d. Two of the five (10 points) e. One of the five (5 points) Total Numer of Points Any score elow C needs remediation! A B C D F 90 points and aove 80 points and aove 70 points and aove 60 points and aove 59 points and elow Johnny Wolfe Jay High School Santa Rosa County
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