Modeling Data With Quadratic Functions §5.1

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

Symmetry! Think back to geometry What does it mean to be symmetrical?

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

Classifying Functions A quadratic function is a function that can be written in the standard form f(x) = ax2 + bx + c where a≠0. Linear or Quadratic? Label each term as a quadratic, linear, or constant term. f(x) = (x2 + 5x) – x2 f(x) = (x – 5)(3x – 1) f(x) = x(x + 3) Label: Quadratic term, Linear term, constant term Key Point: ax2 + bx + c is only a quadratic when a≠0. Problem may start with x2 terms but if there are none when simplified, the expression is linear.

Key Terms Parabola Vertex of a parabola The graph of a quadratic function. The point at which the parabola intersects the axis of symmetry. Axis of symmetry Maximum/Minimum The line that divides a parabola into two parts that are mirror images. The y-value of the vertex of a parabola. “You can fold a parabola so that the two sides match exactly. This is why a line of symmetry exists, or as we call it, an axis of symmetry. Just like every coordinate plane has an x-axis and a y-axis, every parabola has an axis of symmetry. We will learn more about this axis tomorrow, but in a more general sense, it’s just like when you find lines of symmetry in different shapes or objects.” The highest or lowest point of a parabola is a vertex. The vertex is always located on the axis of symmetry.

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 = ax2 + bx +c. The result is a system of three linear equations. x y 2 3 13 4 29

Finding a Quadratic Model Substitute the values of x and y into y = ax2 + bx +c. The result is a system of three linear equations. x y 1 2 -3 3 -10

Real-World Connection Hydraulics 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. Elapsed Time Water Level 0 s 120 mm 10s 100 mm 20 s 83 mm 30 s 66 mm 40 s 50 mm 50 s 37 mm 60 s 28 mm

What are we being asked to do? Model the data Estimate the water level at 35 seconds Enter the data Use QuadReg. Graph the data and the function. Use the table feature to find f(35). Elapsed Time Water Level 0 s 120 mm 10s 100 mm 20 s 83 mm 30 s 66 mm 40 s 50 mm 50 s 37 mm 60 s 28 mm f(35)~58mm

Use your quadratic model to approximate the water level at 25 seconds. Use the quadratic model to predict the water level at 3 minutes. Elapsed Time Water Level 0 s 120 mm 10s 100 mm 20 s 83 mm 30 s 66 mm 40 s 50 mm 50 s 37 mm 60 s 28 mm f(25) ~ 73