Quadratic Functions; Parabolas

Determining if a Function is Quadratic Highest exponent in the equation is 2, no more no less

Concave up or Concave down? Every quadratic function can be described as either concave up or concave down Concave up means the graph has a minimum vertex and opens upwards Concave down means the graph has a maximum vertex and opens downwards ‘a’ in the equation determines the concavity of the graph F(x)= ax 2 + bx + c If ‘a’ is positive, then the graph is concave up If ‘a’ is negative, then the graph is concave down

Examples

Finding the Vertex Point (maximum or minimum value)

Examples

Finding Axis of Symmetry Axis of symmetry is the same as finding the x coordinate of the vertex

Maximizing Revenue Suppose the monthly revenue from the sale of Samsung 42- inch plasma televisions is given by the function: R(x) = -0.1x x dollars, where x is the number of televisions sold Find the vertex and axis of symmetry of the graph. Determine if the vertex represents a maximum or minimum. Interpret the vertex in the content of the problem.

Foreign-Born Population Using data from 1900 through 2004, the percent of the US population that was foreign-born can be modeled by the equation: Y = x 2 – 0.138x , where x is the number of years after 1900 During what year does the model indicate that the percent of foreign-born population was a minimum? What is the minimum percent?

Suppose an object is shot or thrown into the air and then falls. If air resistance is ignored, the height in feet of the object after t seconds can be modeled by: S(t) = -16t 2 + v 0 t + h o Where -16 ft/sec 2 is the acceleration due to gravity v 0 ft/sec is the initial velocity (at t=0) h o is the initial height in feet (at t=0)

Height of a Ball A ball is thrown at 64 feet per second from the top of an 80- foot-high building. Write the quadratic function that models the height(in feet) of the ball as a function of time t(in seconds) Find the vertex of the graph Explain the meaning of the coordinates

Vertex Form of Quadratic Function When a quadratic function is written in the form f(x)=ax 2 +bx+c, we can calculate the coordinates of the vertex. However, if the quadratic function is written in the form y=a(x-h) 2 + k, the vertex of the parabola is at (h,k)

Graph of a Quadratic Function

Minimizing Cost The cost for producing Champions golf hats is given by the function: C(x) = 0.2(x-40) dollars Find the vertex of this function. Is the vertex a minimum or maximum? Interpret vertex in context of problem? Describe what happens to the function between x=0 and the x-coordinate of the vertex. What does this mean in context of the problem?

Profit Right Sports Management had its monthly maximum profit, $450,000, when it produced and sold 5500 Waist Trimmers. Its fixed cost is $155,000. If the profit can be modeled by a quadratic function of x, the number of Waist Trimmers produced and sold each month, find this quadratic function P(x)

Vertex Form of Quadratic Function Write the vertex form of the equation of the quadratic function from the general form y=2x 2 - 8x + 5 by first finding the vertex and a point on the parabola

Graphing a Quadratic Find the x-coordinate of the vertex Make an x/y table with the x coordinate of the vertex in the middle Choose a couple x values below vertex and a couple above Find ordered pairs, plot points, connect

Which Graph has larger value for ‘a’

Homework Pages ,3,5,17,21,23,27,33,35,37,49,51, 53,57,61,67,72,75