3.3 Analyzing Graphs of Quadratic Functions  Find the vertex, the axis of symmetry, and the maximum or minimum value of a quadratic function.  Graph quadratic functions.  Solve applied problems involving maximum and minimum function values.

Quadratic Function A quadratic function is one of the form f(x)=ax 2 +bx+c where a is not zero. The graph of a quadratic function is U-shaped and is called a parabola.

Graph of a Parabola Upward Concavity, a > 0 a.Open upward or downward b.Relative Min or Max Point c.Absolute Min or Max Point d.Domain e.Range f.Increasing g.Decreasing h.Axis of Symmetry i.Vertex j.Find equation from base function

Graph of a Parabola Downward Concavity, a < 0 a.Open upward or downward b.Relative Min or Max Point c.Absolute Min or Max Point d.Domain e.Range f.Increasing g.Decreasing h.Axis of Symmetry i.Vertex j.Find equation from base function

The vertex of a parabola of the form f (x) = ax 2 + bx + c is given by: x-coordinate: y-coordinate: Vertex of a Parabola

f (x) = x 2 + 4x - 12 a) Will the vertex be a maximum or a minimum? b) Find the vertex of the quadratic equation. f (x) = -x 2 - 6x + 3 a) Will the vertex be a maximum or a minimum? b) Find the vertex of the quadratic equation. Example

1. Determine the concavity: If a > 0, the parabola is concave up; if a < 0, the parabola is concave down. 2. Find the vertex. 3. Identify the y-intercept. 4. Identify the x-intercepts. 5. Use symmetric points from axis of symmetry. Sketching the Graph of f (x) = ax 2 + bx + c Sketch the graph of a)y = x 2 + 2x – 24

Sketch graph y =x 2 +2x–24

A quadratic function given in the form f (x) = a(x-h) 2 + k is written in vertex form. (h, k) is the coordinate of the vertex. If a > 0 the parabola is concave up and if a < 0 the parabola is concave down. Vertex Form

f (x) = (x 2 )/2 - 4x + 8 Find the vertex, the axis of symmetry, and the maximum or minimum value (by completing the square) Example

Given a) Is the vertex a minimum or a maximum? b) Determine the coordinates of the vertex. c) Find the equation of the axis of symmetry. Example

Bob sells gadgets for dune buggies. The cost of operating his shop is given by C(x) = 0.08(x - 85) where x is the number of gadgets sold. a) How many gadgets must he sell to minimize his cost? b) State the minimum total cost. Example

Write an equation of a parabola whose vertex is (2, 2) and passes though the point (3, -8). Example

Write an equation of a parabola

A model rocket is launched with an initial velocity of 100 ft./sec from the top of a hill that is 20 ft. high. Its height, in feet, t seconds after it has been launched is given by he function s(t) = -16t t Determine the time at which the rocket reaches its maximum height and find the maximum height. Example

A landscaper has enough stone to enclose a rectangular koi pond next to an existing garden wall of the Eagleman's’ house with 24 ft of stone wall. If the garden wall forms one side of the rectangle, what is the maximum area that the landscaper can enclose? What dimensions of the koi pond will yield this area? Example