GRAPH OF QUADRATIC FUNCTIONS PRAYER ATTENDANCE CLASSROOM MANAGEMENT REVIEW Complete the table of values for x and y. 2 y 2x x -3 -2 -1 0 1 2 3 4 5 18 8 2 8 2 0 18 32 50 y PRE-TEST: Lance, a Grade 9 student is a Sepak Takraw player in the City Meet. In one of his games he kicked the ball forming a 2 f ( x ) x 5. parabolic path with an equation of Draw the graph and determine the following: 1. equation of the axis of symmetry 2. vertex 3. highest / lowest point obtained by the graph 4. direction of the opening of the parabola 5. Opening of the graph MOTIVATION INTRODUCTION TO PARABOLAS MOTIVATIONAL ACTIVITY Given the graph of a quadratic function , can you determine the intercepts, axis of symmetry, vertex, and direction of the opening of the parabola? LESSON PROPER Quadratic Functions The graph of a quadratic function is a: parabola y A parabola can open up or down. Vertex If the parabola opens up, the lowest point is called the vertex (minimum). If the parabola opens down, the vertex is the highest point (maximum). Vertex NOTE: if the parabola opens left or right it is not a function! x Standard Form The standard form of a quadratic function is: y = ax2 + bx + c y The parabola will open up when the a value is positive. a>0 The parabola will open down when the a value is negative. a¹ 0 x a<0 Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. Axisy of Symmetry We call this line the Axis of symmetry. x If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side. The Axis of symmetry ALWAYS passes through the vertex. Finding the Axis of Symmetry When a quadratic function is in standard form y = ax2 + bx + c, b the equation of the Axis of symmetry is x 2a This is best read as … ‘the opposite of b divided by the quantity of 2 times a.’ 2 Find the Axis of symmetry for y = 3x – 18x + 7 a = 3 b = -18 The Axis of symmetry is x= Finding the Vertex The Axis of symmetry always goes through the Vertex Thus, the Axis of symmetry gives _______. X-coordinate of the vertex. us the ____________ Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry b x 2a a = -2 b=8 The xcoordinate of the vertex is 2 Finding the Vertex Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex. y = -2 ( 2 ) +8 ( 2 ) - 3 2 = - 2 ( 4 ) + 16 - 3 = - 8 + 16 - 3 = 5 The vertex is (2 , 5) Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. b STEP 1: Find the Axis of symmetry using: x 2a STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry. Graphing a Quadratic Function Graph : y = 2x - 4x -1 2 y x =1 STEP 1: Find the Axis of symmetry - b 4 x= = =1 2 a 2(2) STEP 2: Find the vertex Substitute in x = 1 to find the y – value of the vertex. 2 y = 2 (1) - 4 (1)- 1 = - 3 x Vertex : (1, - 3) Graphing a Quadratic Function Graph : y = 2x - 4x -1 2 STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. x y 2 –1 3 5 2 y = 2 (2) - 4 (2)- 1 = - 1 2 y = 2 (3) - 4 (3)- 1 = 5 y x Y-intercept of a Quadratic Function y = 2x - 4x -1 2 Y-axis y The y-intercept of a Quadratic function can Be found when x = 0. y = 2x 2 - 4 x - 1 = 2(0) - 4(0) - 1 2 = 0 - 0 -1 = -1 The constant term is always the y- intercept x Solving a Quadratic The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation. The number of real solutions is at most two. No solutions One solution Two solutions X=3 X= -2 or X = 2 Identifying Solutions Find the solutions of 2x - x2 = 0 The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2 The xintercepts(or Zero’s) of f(x)= 2x – x2 are the solutions to 2x - x2 = 0 X = 0 or X = 2 Analyze the graph of a quadratic function , and determine the following: 1. Intercepts (points where the graph passes the x and y axes) X=0, Y=0 2. equation of the axis of symmetry X=0 3. vertex (turning point of the graph) (0,0) 4. highest / lowest point obtained by the graph (0,0) 5. direction of the opening of the parabola UPWARD GUIDED PRACTICE Draw the graph of a quadratic function f ( x) x 2 x 1 and identify the vertex, and opening of the graph. State whether the vertex is a minimum or a maximum point, and write the equation of its axis of symmetry 2 (-1,0) Vertex _______ UPWARD Opening of the graph __ MINIMUM point Vertex is a ___ X=-1 Equation of the axis of symmetry _____ The graph of a quadratic function is called parabola. The parabola opens upward or downward. It has a turning point called vertex which is either the lowest point or the highest point of the graph. If the value of a>0, it opens upward and has a minimum point but if a<0, the parabola opens downward and has a maximum point. There is a line called the axis of symmetry which divides the graph into two parts such that one-half of the graph is a reflection of the other half. If the quadratic function is expressed in the form , the vertex is the point (h,k). The line x = h is the axis of symmetry and k is the minimum or maximum value of the function. Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. b STEP 1: Find the Axis of symmetry using: x 2a STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry. APPLICATION Draw the graph of a quadratic function and determine the vertex, domain, range, and opening of the graph. State whether the vertex is a minimum or a maximum point, and write the equation of its axis of symmetry. 2 GroupsGroup 1 and 12:: f ( x) x 5 x 6 2 GroupsGroup 3 and 24:: f ( x) x 4 x 4 2 Group 3 : Groups 5 and 6: f ( x) x 2 Group 4 : 𝑓 𝑥 = 𝑥 2 − 4𝑥 − 6 EVALUATION: Jaime, a Grade 10 student is a Sepak Takraw player in the City Meet. In one of his games he kicked the ball forming a 2 parabolic path with an equation of f ( x) x 5. Draw the graph and determine the following: 1. equation of the axis of symmetry 2. vertex 3. highest / lowest point obtained by the graph 4. direction of the opening of the parabola 5. Opening of the graph ASSIGNMENT 1. Take a picture on your neighborhood that shows a parabola. 2.Find the vertex, axis of symmetry. THANK YOU