## Presentation on theme: "Quadratic Equations and Functions"— Presentation transcript:

Algebra I Algebra I ~ Chapter 10 ~ Quadratic Equations and Functions Lesson 10-1 Exploring Quadratic Graphs Lesson 10-2 Quadratic Functions Lesson 10-3 Finding & Estimating Square Roots Lesson 10-4 Solving Quadratic Equations Lesson 10-5 Factoring to Solve Quadratic Equations Lesson Completing the Square Lesson Using the Quadratic Formula Lesson Using the Discriminant Lesson Choosing a Linear, Quadratic, or Exponential Model Chapter Review

Lesson 10-1 Cumulative Review Chap 1-9

Lesson 10-1 Notes Quadratic Function – a function in the form ax2 + bx + c, where a ≠ 0. Examples ~ y = 2x2, y = x2 -7, y = x2 – x – 3 The graph of a quadratic function is a parabola… The graph of y = x2 is ~> ~> ~> A parabola can be folded so that the two sides match exactly. The line that divides the parabola into two matched sides is called the axis of symmetry. The highest or lowest point of a parabola is called its vertex. If a > 0 in y = ax2 + bx + c ~> ~> If a < 0 in y = ax2 + bx + c ~> The vertex is identified as an ordered pair and as minimum or maximum… The parabola opens upward The vertex is the minimum point The parabola opens downward The vertex is the maximum point

Lesson 10-1 Notes Identify the vertex of each graph and tell whether it is a minimum or a maximum. Graphing y = ax2 Make a table of values. Graph the points. (3) Find the corresponding points on the other side of the axis of symmetry. Graph f(x) = -2x2 The value of a affects the width of the parabola as well as the direction it opens. You can order quadratic functions by their widths. Order y = x2, y = ½x2, and y = -2x2 from widest to narrowest… Graphing y = ax2 + c The value of c translates the vertex of the graph up (+) or down (-).

Lesson 10-1 Notes Graph y = 2x2 + 3 Graph y = -1/2x2 – 4 In summary… (1) The coefficient of x2, a, determines the width and whether the parabola points upward (+) or downward (-). The constant, c, determines the vertex location above or below 0. (3) Ordering quadratic graphs by width, the smaller the coefficient, a, of x2, the wider the graph.

Lesson 10-1 Exploring Quadratic Graphs Homework Homework – Practice 10-1 #1-26

Quadratic Functions Lesson 10-2 Practice 10-1

Quadratic Functions Lesson 10-2 Practice 10-1

Quadratic Functions Lesson 10-2 Practice 10-1

Notes Quadratic Functions Lesson 10-2 y = 2x2 + 2x
Graphing y = ax2 + bx + c y = 2x2 + 2x (1) Find the axis of symmetry… x = -b/2a Then find the y coordinate. These are the coordinates of the vertex. (2) Find two other points on the graph. (3) Reflect those two points over the axis of symmetry. Draw the parabola. Graph 2x2 + 2x Graph f(x) = x2 – 6x + 9 Graphing Quadratic Inequalities y ≤ x2 + 2x – 5 Graph the boundary curve… Shade the area below the curve because it is less than or equal to.

Quadratic Functions Lesson 10-2 Notes Graph y > x2 + x + 1

Lesson 10-2 Quadratic Functions Homework Homework ~ Practice 10-2 even

Finding & Estimating Square Roots
Lesson 10-3 Practice 10-2

Finding & Estimating Square Roots
Lesson 10-3 Practice 10-2

Finding & Estimating Square Roots
Lesson 10-3 Practice 10-2

Finding & Estimating Square Roots
Lesson 10-3 Practice 10-2

Finding & Estimating Square Roots
Lesson 10-3 Notes Finding Square Roots Every positive number has two square roots… The square root of 16 = 4 and -4 or ± 4. √25 means the positive or principal square root of 25 which is 5. -√25 means the negative square root of 25 which is -5. You can use ± to represent both square roots. Simplifying Square Root Expressions √64 = √100 = ±√49 = √1/25 = √121 = Rational & Irrational Square Roots Rational square roots have a terminating or repeating decimal… Irrational square roots have decimals that do not repeat. ±√81 = √8 = √225 = √75 = ±√1/4 = Estimating Square Roots You can estimate square roots by using perfect squares. Estimation places the square root between two consecutive integers.

Finding & Estimating Square Roots
Lesson 10-3 Notes Example - √18.5 √16 < √18.5 < √25 so… 4 < √18.5 < 5 so √18.5 is between 4 and 5. Your turn -√105 is between what two consecutive integers? Approximating Square Roots with a Calculator Find √18.5 to the nearest hundredth… Find √17.81 to the nearest hundredth Find -√203 to the nearest hundredth

Finding & Estimating Square Roots
Lesson 10-3 Finding & Estimating Square Roots Homework Homework – Practice 10-3 odd

Lesson 10-4 Practice 10-3

Lesson 10-4 Notes Solving Quadratic Equations by Graphing A quadratic equation is an equation that can be written in the form… ax2 + bx+ c = 0, where a ≠ 0. This is the standard form of a quadratic equation. Quadratic equations can have two, one, or no real-number solutions. Algebra I focusing only on real-number solutions. Solve by graphing… x2 – (The solution(s) are the x-intercepts) What about x2 = 0 ? x2 – 1 = 0 2x2 + 4 = 0 x2 – 16 = -16 Solving Quadratic Equations Using Square Roots To solve an equation in the form x2 = a; find the square roots of both sides.

Lesson 10-4 Notes t2 – 25 = 0 t2 = 25 t = ± 5 Find the solution(s) 3n = 12 Try… g = 0 Factoring can also be used to solve the quadratic equation… x2 - 9 = 0 (x + 3) (x – 3) = 0 (x + 3) = 0 or (x – 3) = 0 x = x = Solutions ±3

Lesson 10-4 Homework Homework – Practice 10-4 odd

Lesson 10-5 Practice 10-4

~ Chapter 10 ~ Algebra I Algebra I Chapter Review

Chapter 10 Review Part 1 Chapter Review

Chapter 10 Review Part 1 Chapter Review

Chapter 10 Review Part 1 Chapter Review

Lesson 10-5 Notes Using the Zero-Product Property If ab = 0, then a = 0 or b = 0 Solve (x + 7) (x – 4) = 0 So… x + 7 = 0 or x – 4 = 0 x = or x = 4 Your turn… Solve (3y – 5) (y – 2) = 0 Solving by Factoring x2 – 8 x – 48 = 0 (x – 12) (x + 4) = 0 x – 12 = 0 or x + 4 = 0 x = or x = -4 Your turn… x2 + x – 12 = 0 x = -4 or x = 3

Lesson 10-5 Notes 2x2 – 5x = 88 2x2 – 5x – 88 = 0 Your turn… Solve x2 – 12x = -36 x2 – 12x + 36 = 0 (x - 6)2 = 0 x = 6

Lesson 10-5 Homework ~ Homework ~ Practice 10-5 even

Completing the Square Lesson 10-7 Practice 10-5

Lesson 10-7 Notes Using the Quadratic Formula If ax2 + bx + c = 0, and a ≠ 0, then… x = -b ± b2 – 4ac 2a Make sure your quadratic equation is in standard form… Solve x2 + 6 = 5x x2 – 5x + 6 = 0 x = - (-5) ± (-5)2 – 4(1)(6) 2(1) x = 5 ± – 24 2 x = 5 ± √ 1 = or 5 – 1 = or = or 2

Lesson 10-7 Notes Your turn… Solve using the quadratic formula x2 – 4x = 117 x2 – 2x – 8 = 0 Finding Approximate Solutions 2x2 + 4x – 7 = 0 x = - (4) ± (4)2 – 4(2)(-7) = -4 ± – (-56) 2(2) 4 x = -4 + √72 or √72 ≈ or ≈ or -3.12 7x2 – 2x – 8 = 0

Lesson 10-7 Homework Homework – Practice 10-7 even

Using the Discriminant
Lesson 10-8 Practice 10-7

Using the Discriminant
Lesson 10-8 Notes Number of Real Solutions of a Quadratic Equation Discriminant – The expression under the radical in the quadratic formula (b2 – 4ac) The discriminant can be used to determine how many solutions a quadratic equation has before you solve it… If b2 – 4ac > 0, there are 2 solutions If b2 – 4ac = 0, there is 1 solution If b2 – 4ac < 0, there are no solutions Using the Discriminant Find the number of solutions for x2 – 2x – 3 b2 – 4ac = (-2)2 – 4(1)(-3) 4 – (-12) = = 16 > 0 , so there are 2 solutions. Your turn… Find the number of solutions for 3x2 – 4x – 7 Find the number of solutions for 5x2 + 8 = 2x

Using the Discriminant
Lesson 10-8 Homework Homework – Practice 10-8 odd & Chapter 10 Review Part 2

Using the Discriminant
Lesson 10-8 Homework

~ Chapter 10 ~ Algebra I Algebra I Chapter Review

~ Chapter 10 ~ Algebra I Algebra I Chapter Review