2 NOTES: A quadratic function standard form y = ax2 + bx + c, a ≠ 0. Quadratic parent functiony = x2OR y = 1x2 + 0x + 0A parabola opens upward when a > 0.A parabola opens downward when a < 0.
3 a>0 (Positive a’s) a<0 (Negative a’s) Opens upward L9-1 NOTES:Identifying Quadratic Functions1) standard form y = ax2 + bx + c2) It has constant second differences3) The function graphs as a parabola.Vertex: The highest or lowest point on a parabola.a>0 (Positive a’s)a<0 (Negative a’s)Opens upwardOpens downwardVertex is a low ptVertex is a high pt
4 Lesson Quiz: Part I1. Is y = –x – 1 quadratic? Explain.2. Graph y = 1.5x2.No; there is no x2-term, so a = 0.
5 Lesson Quiz: Part IIUse the graph for Problems 3-5.3. Identify the vertex.4. Does the function have a minimum or maximum? What is it?5. Find the domain and range.(5, –4)max; –4D: all real numbers;R: y ≤ –4
6 L 9-2 NOTESA zero of a function: an x-value that makes y=0.( x-intercept(s) of the quadratic function).A quadratic function may have one, two, or no zeros.Axis of symmetry: A vertical line that divides a parabola into two symmetrical halves.-it always passes through the vertex.
7 x-coordinate of the vertex. So, x = 3 Pg 600NOTESOne ZeroTwo ZerosVertex(3, 0)(-4, 0) and (0, 0)Axis of symmetryx-coordinate of the vertex. So, x = 3Average the 2 zeros:(-4+0)/2 = -2 so x=-2No Zeros: use the formula (works for all quads)Pg 601
8 Once you have found the axis of symmetry, you can use it to identify the vertex. Pg 601
9 Lesson Quiz: Part I1. Find the zeros and the axis of symmetry of the parabola.2. Find the axis of symmetry and the vertex of the graph of y = 3x2 + 12x + 8.zeros: –6, 2; x = –2x = –2; (–2, –4)
10 Lesson Quiz: Part II3. The graph of f(x) = –0.01x2 + x can be used to model the height in feet of a curved arch support for a bridge, where the x-axis represents the water level and x represents the distance in feet from where the arch support enters the water. Find the height of the highest point of the bridge.25 feet
11 NOTESFor Quadratic y = ax2 + bx + c, the y-intercept is c.Graphing:Find the axis of symmetry x=Find the vertex: sub x into the function, solve for yGraph the y-intercept and reflect across axis of symmetry.Pick a couple more x’s and solve for the y’s, graph and reflect.
12 Lesson Quiz1. Graph y = –2x2 – 8x + 4.2. The height in feet of a fireworks shell can be modeled by h(t) = –16t t, where t is the time in seconds after it is fired. Find the maximum height of the shell, the time it takes to reach its maximum height, and length of time the shell is in the air.784 ft; 7 s; 14 s