Math (L-3a) Learning Targets: I can find the vertex from intercept solutions calculated by quadratic formula. PART I: Emmett s teacher asked him to analyze the table of values of a quadratic function to find key features. The table of values is shown below: x y -4 1-3 1-5 -1 0 0-3 1-4 -3 3 0 4 5 1. Emmett quickly determines that the y-intercept is at the point (0, -3) and the x-intercepts are at (-1, 0) and (3, 0). Is this correct? Explain.. Emmett would like to determine the axis of symmetry and identify the vertex of the parabola. He notices the following x values correspond to the same y values: 0 and, -1 and 3, - and 4. He believes that the axis of symmetry should be the same distance away from these x values in these sets, so he averages these sets to find the point in the middle: 0+ = 1, 1+3 = 1, +4 1. Each time, the result was 1, and Emmett determines the axis of symmetry lies at x = 1. Then, he finds the vertex to be at (1, -4). Is Emmett s conclusion correct? Explain. =
Math (L-3a) PART II: Emmett s teacher now presents him with an equation of a quadratic function in standard form and asks him to identify the key features. The quadratic function is y = x + 5x + 6. 1. Emmett immediately identifies the y-intercept at the point (0, 6). How did he determine the y- intercept so quickly?. Emmett knows the x-intercepts have y values of 0, so he replaces the y with a 0; 0 = x + 5x + 6. He decides to solve the equation by factoring. 0 = (x + )(x + 3). Emmett concludes that the x- intercepts lie at the points (, 0) and (3, 0). Is he correct? Explain. 3. What is another method Emmett could have used to find the x-intercepts from standard form? 4. Use the method in #3 to check the x-intercepts from #. 5. Now that the x-intercepts have been identified, Emmett can average the x values to determine the x-coordinate of the vertex. What is the x-coordinate of the vertex? 4. How can Emmett find the y-coordinate of the vertex?
Math (L-3a) 5. What is the vertex of the parabola? 6. Graph the quadratic function y = x + 5x + 6 from the key features found. PART III: Emmett utilized his knowledge of symmetry to determine if he knows the x-intercepts of a quadratic function that he can find the x-coordinate of the vertex by averaging the x-intercepts. When given the function in standard form, he factored to find the x-intercepts. The Quadratic Formula can also be used to find the x-intercepts when the function is represented in standard form. 1. Use this knowledge of symmetry and the Quadratic Formula solutions to find a general rule for identifying the vertex of any quadratic function in standard form: y = ax + bx + c where a 0. PART IV: Use the rule for finding the vertex of any quadratic function in standard form from PART III to find the vertex of the following quadratics in standard form. 1. y = x + 6x + 4. y = -x - 4x 4 3. y = x + 9 4. y = x + 4x + 1 5. y = x + 5x -
Math (L-3a) Answer Key PART I: Emmett s teacher asked him to analyze the table of values of a quadratic function to find key features. The table of values is shown below: x y -4 1-3 1-5 -1 0 0-3 1-4 -3 3 0 4 5 1. Emmett quickly determines that the y-intercept is at the point (0, -3) and the x-intercepts are at (-1, 0) and (3, 0). Is this correct? Explain. Yes. He looked at the table and identified the place where the x value was 0 to find where the graph would cross the y-axis, (0, 3). He also identified the points with y values of 0 as the points where the graph would cross the x-axis (-1, 0) and (3, 0). A quadratic function has a maximum of x-intercepts and 1 y-intercept, so these were all of the intercepts.. Emmett would like to determine the axis of symmetry and identify the vertex of the parabola. He notices the following x values correspond to the same y values: 0 and, -1 and 3, - and 4. He believes that the axis of symmetry should be the same distance away from these x values in these sets, so he averages these sets to find the point in the middle: 0+ = 1, 1+3 = 1, +4 1. Each time, the result was 1, and Emmett determines the axis of symmetry lies at x = 1. Then, he finds the vertex to be at (1, -4). Is Emmett s conclusion correct? Explain. Yes. The axis of symmetry should lie directly between symmetric points on the parabola. It should be equidistant from the points. To find the x-coordinate of the midpoint (in this case, the vertex), simply calculate an average of the two points x-coordinates. The y-value may be found from the table. =
Math (L-3a) PART II: Emmett s teacher now presents him with an equation of a quadratic function in standard form and asks him to identify the key features. The quadratic function is y = x + 5x + 6. 1. Emmett immediately identifies the y-intercept at the point (0, 6). How did he determine the y- intercept so quickly? The y-intercept is the point in which the graph crosses the y-axis, and the x-coordinate is 0. Thus, he could quickly substitute 0 in for x, and simplify. He may have also remembered that doing this in standard form will always give a y-intercept equal to the constant, c.. Emmett knows the x-intercepts have y values of 0, so he replaces the y with a 0; 0 = x + 5x + 6. He decides to solve the equation by factoring. 0 = (x + )(x + 3). Emmett concludes that the x- intercepts lie at the points (, 0) and (3, 0). Is he correct? Explain. No. The x values that satisfy 0 = x + and 0 = x + 3 are x = - and x = -3 respectively. Thus, the points in which the x-intercepts lie are (-, 0) and (-3, 0). 3. What is another method Emmett could have used to find the x-intercepts from standard form? The Quadratic Formula; x = b± b 4ac. a 4. Use the method in #3 to check the x-intercepts from #. x = 5± ( 5) 4(1)(6) (1) = 5±1 = 5+1 or 5 1 = - or -3. 5. Now that the x-intercepts have been identified, Emmett can average the x values to determine the x-coordinate of the vertex. What is the x-coordinate of the vertex? x = +( 3) = 5 4. How can Emmett find the y-coordinate of the vertex? Replace the x value in function with 5, and simplify to find the y value. In other words, find f( 5 ). 5. What is the vertex of the parabola? y = ( 5 ) + 5 ( 5 ) + 6 = 1 4
Math (L-3a) 6. Graph the quadratic function y = x + 5x + 6 from the key features found. ( 5, 1 4 ) PART III: Emmett utilized his knowledge of symmetry to determine if he knows the x-intercepts of a quadratic function that he can find the x-coordinate of the vertex by averaging the x-intercepts. When given the function in standard form, he factored to find the x-intercepts. The Quadratic Formula can also be used to find the x-intercepts when the function is represented in standard form. 1. Use this knowledge of symmetry and the Quadratic Formula solutions to find a general rule for identifying the vertex of any quadratic function in standard form: y = ax + bx + c where a 0. The Quadratic Formula; x intercepts = b± b 4ac. Average these x values to find the x- a coordinate of the vertex as follows: x = ( b, f ( b )). z a b+ b 4ac + b b 4ac a a = b b a = a = b a. Thus, the vertex is PART IV: Use the rule for finding the vertex of any quadratic function in standard form from PART III to find the vertex of the following quadratics in standard form. 1. y = x + 6x + 4 (-3, -5). y = -x - 4x 4 (-, 0) 3. y = x + 9 (0, 9) 4. y = x + 4x + 1 (-1, -1) 5. y = x + 5x - ( 5, 33 4 )