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3.1Quadratic Functions and Models 3.2Quadratic Equations and Problem Solving 3.3Quadratic Inequalities 3.4Transformations of Graphs Quadratic Functions and Equations 3

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quadratic Functions and Models ♦ Learn basic concepts about quadratic functions and their graphs. ♦ Complete the square and apply the vertex formula. ♦ Graph a quadratic function by hand. ♦ Solve applications and model data. 3.1

Slide 3- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Basic Concepts Recall that a linear function can be written as f(x) = ax + b (or f(x) = mx + b). The formula for a quadratic function is different from that of a linear function because it contains an x 2 term. f(x) = 3x 2 + 3x + 5g(x) = 5  x 2

Slide 3- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quadratic Function The graph of a quadratic function is a parabola—a U shaped graph that opens either upward or downward. A parabola opens upward if a is positive and opens downward if a is negative. The highest point on a parabola that opens downward and the lowest point on a parabola that opens upward is called the vertex. The vertical line passing through the vertex is called the axis of symmetry. The leading coefficient a controls the width of the parabola. Larger values of |a| result in a narrower parabola, and smaller values of |a| result in a wider parabola.

Slide 3- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples of different parabolas

Slide 3- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the graph of the quadratic function shown to determine the sign of the leading coefficient, its vertex, and the equation of the axis of symmetry.

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Slide 3- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write the formula f(x) = x x + 23 in vertex form by completing the square. Solution

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the vertex of the graph of.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the vertex formula to write f(x) =  3x 2  3x + 1 in vertex form.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph the quadratic equation g(x) = x 2  1.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Applications and Models Example A junior horticulture class decides to enclose a rectangular garden, using a side of the greenhouse as one side of the rectangle. If the class has 32 feet of fence, find the dimensions of the rectangle that give the maximum area for the garden. L W

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A model rocket is launched with an initial velocity of v o = 150 feet per second and leaves the platform with an initial height of h o = 10 feet. a) Write a formula s(t) that models the height of the rocket after t seconds. b) How high is the rocket after 3 seconds? c) Find the maximum height of the baseball. Support your answer graphically. Solution a)

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quadratic Equations and Problem Solving ♦ Understand basic concepts about quadratic equations ♦ Use factoring, the square root property, completing the square, and the quadratic formula to solve quadratic equations ♦ Understand the discriminant ♦ Solve problems involving quadratic equations 3.2

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Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Quadratic Equations The are four basic symbolic strategies in which quadratic equations can be solved. Factoring Square root property Completing the square Quadratic formula

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factoring A common technique used to solve equations that is based on the zero-product property. Example

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Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A rescue helicopter hovers 68 feet above a jet ski in distress and drops a life raft. The height in feet of the raft above the water is given by Determine how long it will take for the raft to hit the water after being dropped from the helicopter. Solution The raft will hit the water when its height is 0 feet above the water.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Completing the Square Completing the square is useful when solving quadratic equations that do not factor easily. If a quadratic equation can be written in the form where k and d are constants, then the equation can be solved using

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve 2x 2 + 6x = 7.

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Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve the equation

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Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the discriminant to determine the number of solutions to the quadratic equation

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A box is is being constructed by cutting 2 inch squares from the corners of a rectangular sheet of metal that is 10 inches longer than it is wide. If the box has a volume of 238 cubic inches, find the dimensions of the metal sheet..

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quadratic Inequalities ♦ Solve quadratic inequalities graphically ♦ Solve quadratic inequalities symbolically 3.3

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quadratic Inequalities If an equals sign is replaced by >, , <, or ≤, a quadratic inequality results. A first step in solving a quadratic inequality is to determine the x-values where equality occurs. These x-values are the boundary numbers.

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Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve the inequality. Write the solution set for each in interval notation. a) 3x 2 + x  4 = 0b) 3x 2 + x  4 < 0 c) 3x 2 + x  4 > 0 Solution a) Factoring (3x + 4)(x – 1) = 0 x = – 4/3 x = 1 The solutions are – 4/3 and 1.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued b) 3x 2 + x  4 < 0 c) 3x 2 + x  4 > 0 y < 0 y > 0

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Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve x 2 > 7x – 10 symbolically. Write the solutions in interval notation. Solution Step 1: Rewrite the inequality as x 2 – 7x + 10 > 0 Step 2: Solve x 2 – 7x + 10 = 0 (x – 5)(x – 2) = 0 x = 5 or x = 2 Step 3: These two boundary numbers separate the number line into three disjoint intervals. (– , 2), (2, 5), and (5,  )

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