College Algebra Chapter 3 Polynomial and Rational Functions Section 3.1 Quadratic Functions and Applications Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Graph a Quadratic Function Written in Vertex Form Write f(x) in Vertex Form Find the Vertex of a Parabola by Using the Vertex Formula Solve Applications Involving Quadratic Functions Create Quadratic Models Using Regression

Concept 1 Graph a Quadratic Function Written in Vertex Form

Graph a Quadratic Function Written in Vertex Form (1 of 2) The graph of the quadratic function The parabola opens upward for a > 0 and opens downward for a < 0

Graph a Quadratic Function Written in Vertex Form (2 of 2) The function may be written in vertex form as The vertex is (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.

Skill Practice 1 Repeat Example 1 with

Concept 2 Write f(x) in Vertex Form

Example 3 Complete the square within parentheses. Remove –25 from within the parentheses along with a factor of 2.

Example 4 (1 of 2) Write the given f(x) in vertex form and graph the function. Identify the vertex, x- and y-intercepts, and axis of symmetry.

Example 4 (2 of 2) Write the domain and range in interval notation.

Skill Practice 2 Repeat Example 2 with

Write f(x) in Vertex Form

Concept 3 Find the Vertex of a Parabola by Using the Vertex Formula

Find the Vertex of a Parabola by Using the Vertex Formula

Example 5 (1 of 2) Given f(x) determine the vertex by using the vertex formula and by writing in vertex form.

Skill Practice 3 Repeat Example 3 with

Concept 4 Solve Applications Involving Quadratic Functions

Example 6 (1 of 3) 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 f(x) where x is the distance in inches along the ground from Sherlock.

Example 6 (2 of 3) What is the maximum height the bee reaches as it flies from A to B? How far away from Sherlock are flowers A and B?

Skill Practice 4 A quarterback throws a football with an initial velocity of 72 ft/sec at an angle of 25° The height of the ball can be modeled by h(t) where h(t) is the height (in ft) and t is the time in seconds after release. Determine the time at which the ball will be at its maximum height. Determine the maximum height of the ball. Determine the amount of time required for the ball to reach the receiver’s hands if the receiver catches the ball at a point 3 ft off the ground.

Skill Practice 5 A farmer has 200 ft of fencing and wants to build three adjacent rectangular corrals. Determine the dimensions that should be used to maximize the area, and find the area of each individual corral.

Concept 5 Create Quadratic Models Using Regression

Example 7 (1 of 8) 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 (2 of 8) Use the STAT button, then EDIT to enter the x and y data in two lists. Exit this screen.

Example 7 (3 of 8) Use the STAT button, then CALC, choose 5:QuadReg.

Example 7 (4 of 8) Hit ENTER. Then CALCULATE. The equation is:

Example 7 (5 of 8) To see the data and the line graphed: Above the y = key, select STATPLOT Turn Plot1 ON and Select the type of graph.

Example 7 (6 of 8) Graph

Example 7 (7 of 8) Enter y into the equation editor and see the parabola graphed.

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

Skill Practice 6 The funding f(t) (in $ millions) for a drug rehabilitation center is given in the table for selected years t. t 3 6 9 12 15 f(t) 3.5 2.2 2.1 4.9 8 Use regression to find a quadratic function to model the data. During which year is the funding the least? Round to the nearest year. What is the minimum yearly amount of funding received? Round to the nearest million.