+ Modeling Data With Quadratic Functions §5.1

+ Objectives Identify quadratic functions and graphs. Model data with quadratic functions. Graph quadratic functions. By the end of today, you should be able to…

+ Symmetry!

+ Quadratic functions y = x 2 – 3x + 5 y = -x 2 – 1 y = 3x²+12x+5

+ Classifying Functions A quadratic function is a function that can be written in the standard form f(x) = ax 2 + bx + c where a≠0. Linear or Quadratic? Label each term as a quadratic, linear, or constant term. f(x) = (x 2 + 5x) – x 2 f(x) = (x – 5)(3x – 1) f(x) = x(x + 3)

+ Key Terms Parabola The graph of a quadratic function. Axis of symmetry The line that divides a parabola into two parts that are mirror images. Vertex of a parabola The point at which the parabola intersects the axis of symmetry. Maximum/Minimum The y-value of the vertex of a parabola.

+ Points on a Parabola The vertex is ________. The axis of symmetry is ________, the vertical line passing through the vertex. P(1, 2) is one unit to the left of the axis of symmetry. Corresponding point ________ is one unit to the right of the axis of symmetry. Q(0, 8) is two units to the left of the axis of symmetry. Corresponding point ________ is two units to the right of the axis of symmetry.

+ Points on a Parabola The vertex is ________. The axis of symmetry is ________, the vertical line passing through the vertex. P(-2, 1) is one unit to the left of the axis of symmetry. Corresponding point ________ is one unit to the right of the axis of symmetry. Q(1, -2) is two units to the left of the axis of symmetry. Corresponding point ________ is two units to the right of the axis of symmetry.

+ Finding a Quadratic Model Substitute the values of x and y into y = ax 2 + bx +c. The result is a system of three linear equations. xy

+ Finding a Quadratic Model Substitute the values of x and y into y = ax 2 + bx +c. The result is a system of three linear equations. xy

+ Real-World Connection The table shows the height of the water in a cooler as it drains from its container. Model the data with a quadratic function. Use the model to estimate the water level at 35 seconds. Hydraulics Elapsed TimeWater Level 0 s120 mm 10s100 mm 20 s83 mm 30 s66 mm 40 s50 mm 50 s37 mm 60 s28 mm

+ What are we being asked to do? 1. Enter the data 2. Use QuadReg. 3. Graph the data and the function. 4. Use the table feature to find f(35). Model the data Estimate the water level at 35 seconds Elapsed TimeWater Level 0 s120 mm 10s100 mm 20 s83 mm 30 s66 mm 40 s50 mm 50 s37 mm 60 s28 mm

+ a. Use your quadratic model to approximate the water level at 25 seconds. b. Use the quadratic model to predict the water level at 3 minutes. Elapsed TimeWater Level 0 s120 mm 10s100 mm 20 s83 mm 30 s66 mm 40 s50 mm 50 s37 mm 60 s28 mm

+ Homework p (1-14 even, even) #10, 12, 14 – list x-intercept(s) and y-intercept