Quadratic Functions and Models ♦ ♦ Learn basic concepts about quadratic functions and their graphs. ♦ Complete the square and apply the vertex formula. ♦ Graph a quadratic function by hand. ♦ Solve applications and model data. 2.1

Basic Concepts REVIEW: A linear function can be written as f(x) = ax + b (or f(x) = mx + b). The formula for a quadratic function is different from that of a linear function because it contains an x 2 term. f(x) = 3x 2 + 3x + 5g(x) = 5  x 2 A quadratic function is nonlinear. Leading Coefficient

Quadratic Function Properties The graph of a quadratic function is a parabola—a U shaped graph that opens either upward or downward. A parabola opens upward if its leading coefficient a is positive and opens downward if a is negative. The highest point on a parabola that opens downward and the lowest point on a parabola that opens upward is called the vertex. (The graph of a parabola changes shape at the vertex.) The vertical line passing through the vertex is called the axis of symmetry. The leading coefficient a controls the width of the parabola. Larger values of |a| result in a narrower parabola, and smaller values of |a| result in a wider parabola. Leading Coefficient

Examples of different parabolas

Example Use the graph of the quadratic function shown to determine the sign of the leading coefficient, its vertex, and the equation of the axis of symmetry. Leading coefficient: The graph opens downward, so the leading coefficient a is negative. Vertex: The vertex is the highest point on the graph and is located at (1, 3). Axis of symmetry: Vertical line through the vertex with equation x = 1.

The quadratic function f(x) =ax 2 + bx + c can be written in an alternate form that relies on the vertex (h, k). Example: f(x) = 3(x - 4) is in vertex form with vertex (h, k) = (4, 6). What is the vertex of the parabola given by ? f(x) = 7(x + 2) 2 – 9 ? Vertex = (-2,-9)

Example Write the formula f(x) = x x + 23 in vertex form by completing the square. Given formula Subtract 23 from each side. Add (10/2) 2 = 25 to both sides. Factor perfect square trinomial. Subtract 2 form both sides.. Vertex is h= -5 k = -2.What is the vertex?

Example A well-conditioned athlete’s heart rate can reach 200 beats per minute (bpm) during a strenuous workout. Upon stopping the activity, a typical heart rate decreases rapidly and then gradually levels off. See the data table. Graphing this data we get Determine the equation of a parabola that models this data. f(x) = a(x – h) 2 + k How do we start?

From the description, we will take that the minimum heart rate is 80 beats per minute and that occurs when x = 8. That means the vertex is at (8, 80). So using the vertex form we have f(x) = a(x – 8) How do we find the value of coefficient a? a = 1.875

Assignment: Pages 132 – 133 Problems: 18 – 32e; 44 – 52e