Graphing Quadratics

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  • 1.Graphing Quadratic Functions MA.912.A.2.6 Identify and graph functions MA.912.A.7.6 Identify the axis of symmetry, vertex, domain, range, and intercept(s) for a given parabola

2. Quadratic Function y = ax2 + bx + c Quadratic Term Linear Term Constant Term What is the linear term of y = 4x2 3? 0x What is the linear term of y = x2 - 5x ? -5x What is the constant term of y = x2 5x? 0 Can the quadratic term be zero? No! 3. Quadratic Functions The graph of a quadratic function is a: A parabola can open up or down. 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). NOTE: if the parabola opens left or right it is not a function! y x Vertex Vertex parabola 4. y = ax2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Standard Form y x The standard form of a quadratic function is: a > 0 a < 0 a0 5. y x Axis of Symmetry Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the Axis of symmetry. The Axis of symmetry ALWAYS passes through the vertex. If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side. 6. Find the Axis of symmetry for y = 3x2 18x + 7 Finding the Axis of Symmetry When a quadratic function is in standard form the equation of the Axis of symmetry is y = ax2 + bx + c, 2 b a x = This is best read as the opposite of b divided by the quantity of 2 times a. ( ) 18 2 3 x= 18 6 = 3= The Axis of symmetry is x = 3. a = 3 b = -18 7. Finding the Vertex The Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex. STEP 1: Find the Axis of symmetry Vertex Find the vertex of y = -2x2 + 8x - 3 2 b a x = a = -2 b = 8 x= 8 2(2) = 8 4 =2 X-coordinate The x- coordinate of the vertex is 2 8. Finding the Vertex 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. 8 8 2 2 2( 2) 4 b a x = = = = The vertex is (2 , 5) Find the vertex of y = -2x2 + 8x - 3 y = 2 2( ) 2 +8 2( ) 3 = 2 4( )+16 3 = 8+16 3 = 5 9. Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. STEP 1: Find the Axis of symmetry using: 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. 2 b a x = 10. STEP 1: Find the Axis of symmetry ( ) 4 1 2 2 2 b x a - = = = y x Graph:y=2x2 4x1 Graphing a Quadratic Function STEP 2: Find the vertex Substitute in x = 1 to find the y value of the vertex. ( ) ( )2 2 1 4 1 1 3y = - - =- Vertex:1,3( ) x=1 11. 5 1 ( ) ( )2 2 3 4 3 1 5y = - - = STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. y x ( ) ( )2 2 2 4 2 1 1y = - - =- 3 2 yx Graphing a Quadratic Function Graph:y=2x2 4x1 12. y x Y-intercept of a Quadratic Function y=2x2 4x1Y-axis The y-intercept of a Quadratic function can Be found when x = 0. y =2x2 4x 1 =2 0( )2 4(0) 1 =0 0 1 = 1 The constant term is always the y- intercept 13. The number of real solutions is at most two. Solving a Quadratic No solutions 6 4 2 -2 5 f x()= x2-2 x( )+5 6 4 2 -2 5 2 -2 -4 -5 5 One solution X = 3 Two solutions The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation. 14. Identifying Solutions 4 2 -2 -4 5 X = 0 or X = 2 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 x- intercepts(or Zeros) of f(x)= 2x x2 are the solutions to 2x - x2 = 0