MAT 150 Unit 2-1: Quadratic Functions; Parabolas

Objectives  Determine if a function is quadratic  Determine if the graph of a quadratic function is a parabola that opens down  Determine if the vertex of the graph of a quadratic function is a maximum or a minimum  Determine if a function increases or decreases over a given interval  Find the vertex of the graph of a quadratic function  Graph a quadratic function  Write the equation of a quadratic function given information about its graph  Find the vertex form of the equation of a quadratic function

Graph of a Quadratic Function The general form of a quadratic function is f(x) = ax 2 + bx + c where a, b, and c are real numbers with a ≠ 0. The graph of the quadratic function f(x) = ax 2 + bx + c is a parabola with a “turning point” called the vertex. The parabola opens upward (is concave up) if a is positive and the vertex is a minimum point. The parabola opens downward (is concave down) if a is negative and the vertex is a maximum point.

Increasing and Decreasing Functions A function f is increasing on an interval if, for any x 1 and x 2 in the interval, when x 2 > x 1, it is true that f(x 2 ) > f(x 1 ). A function f is decreasing on an interval if, for any x 1 and x 2 in the interval, when x 2 > x 1, it is true that f(x 2 ) < f (x 1 ).

Graphing a Complete Quadratic Function Vertex X-intercepts Y-intercepts

Example Find the vertex and graph the quadratic function f(x) = –2x 2 – 4x + 6. Solution

Example (cont) Find the vertex and graph the quadratic function f(x) = –2x 2 – 4x + 6. Solution The x-intercepts can be found by setting f(x) = 0 and solving for x:

Example (cont) Solution The y-intercepts can be found by computing f(0).

Example—Spreadsheet solution The graph of the function f(x) = –2x 2 – 4x + 6 using Excel.

Example A ball is thrown upward at 64 feet per second from the top of an 60-foot-high building. a. Write the quadratic function that models the height (in feet) of the ball as a function of the time t (in seconds). Solution

Example (cont) b. Find the t-coordinate and S-coordinate of the vertex of the graph of this quadratic function. Solution

Example (cont) c. Graph the model. Solution

Example (cont) d. Explain the meaning of the coordinates of the vertex for this model. Solution

Using the Vertex in a Real-Life Situation

a) How many televisions must be sold to make a maximum amount of revenue? b) What is the maximum amount of revenue to be made?

c) For what x-values is the function increasing? decreasing? What does this mean in the context of the application?

Graph of a Quadratic Function In general, the graph of the function y = a(x  h) 2 + k is a parabola with its vertex at the point (h, k). The parabola opens upward if a > 0, and the vertex is a minimum. The parabola opens downward if a < 0, and the vertex is a maximum. The axis of symmetry of the parabola has equation x = h. The a is the same as the leading coefficient in y = ax 2 + bx + c, so the larger the value of |a|, the narrower the parabola will be.

Example

Write the equation of the quadratic function whose graph is shown. X Y (3,5) (1,1)

Example 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). Solution

Example (cont) Solution

Example Write the vertex form of the equation of the quadratic function from the general form y = 2x 2 – 8x + 3. Solution