Quadratic Functions and Applications

1. Graph a Quadratic Function Written in Vertex Form 2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form 3. Find the Vertex of a Parabola by Using the Vertex Formula 4. Solve Applications Involving Quadratic Functions 5. Create Quadratic Models Using Regression

Graph a Quadratic Function Written in Vertex Form The graph of the quadratic function is a parabola. a < 0 a > 0 The parabola opens upward for a > 0 and opens downward for a < 0

Graph a Quadratic Function Written in Vertex Form The function may be written in vertex form as The vertex is (h, k). a < 0 a > 0 vertex (h, k) axis of symmetry x = h

Example 1: Graph the quadratic function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

Example 2: Graph the quadratic function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

1. Graph a Quadratic Function Written in Vertex Form 2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form 3. Find the Vertex of a Parabola by Using the Vertex Formula 4. Solve Applications Involving Quadratic Functions 5. Create Quadratic Models Using Regression

Use the technique of completing the square to write Example 3: Use the technique of completing the square to write into vertex form and identify the vertex. Factor out the leading coefficient of the x2 term from the two terms containing x. Complete the square within parentheses. Remove –25 from within the parentheses along with a factor of 2.

Example 4: Write in vertex form and graph the function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

Example 4 continued: Write the domain and range in interval notation.

Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form Given a quadratic function defined by

1. Graph a Quadratic Function Written in Vertex Form 2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form 3. Find the Vertex of a Parabola by Using the Vertex Formula 4. Solve Applications Involving Quadratic Functions 5. Create Quadratic Models Using Regression

Find the Vertex of a Parabola by Using the Vertex Formula the x-coordinate of the vertex is given by . To find the y-coordinate, evaluate . The vertex is given by .

Example 5: Given determine the vertex by using the vertex formula and by writing in vertex form.

Example 5 continued:

1. Graph a Quadratic Function Written in Vertex Form 2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form 3. Find the Vertex of a Parabola by Using the Vertex Formula 4. Solve Applications Involving Quadratic Functions 5. Create Quadratic Models Using Regression

Example 6: At the end of his career, Sherlock Holmes retired to small farm in the country and took up a hobby of beekeeping. Being of an analytical mind, he could not just watch a bee fly from flower to flower, he formulated a mathematic model for the parabolic path the bee follows from flower A to flower B. The path is given by where x is the distance in inches along the ground from Sherlock.

Example 6 continued: a. What is the maximum height the bee reaches as it flies from A to B?

Example 6 continued: b. How far away from Sherlock are flowers A and B?

1. Graph a Quadratic Function Written in Vertex Form 2. Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form 3. Find the Vertex of a Parabola by Using the Vertex Formula 4. Solve Applications Involving Quadratic Functions 5. Create Quadratic Models Using Regression

Example 7: Use regression to find a quadratic function to model the data. Round the coefficients to 3 decimal places. x 8 10 12 18 20 23 40 55 60 65 70 75 f (x) 5 13 30 36 35 4

Example 7 continued: 1. Use the STAT button, then EDIT to enter the x and y data in two lists. Exit this screen.

Example 7 continued: 2. Use the STAT button, then CALC, choose 5:QuadReg.

Example 7 continued: 3. Hit ENTER. Then CALCULATE. The equation is:

Example 7 continued: 4. To see the data and the line graphed: Above the y = key, Turn Plot1 ON and select STATPLOT. select the type of graph.

Example 7 continued: Graph

Example 7 continued: 5. Enter into the equation editor and see the parabola graphed.

Example 7 continued: x 8 10 12 18 20 23 40 55 60 65 70 75 f (x) 5 13 30 36 35 4