~ Chapter 10 ~ Quadratic Equations and Functions Algebra I Lesson 10-1 Exploring Quadratic Graphs...

~ Chapter 10 ~Quadratic Equations and

Functions

Algebra I

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 10-6 Completing the Square

Lesson 10-7 Using the Quadratic Formula

Lesson 10-8 Using the Discriminant

Lesson 10-9 Choosing a Linear, Quadratic, or Exponential Model

Chapter Review

Algebra I

Exploring Quadratic Graphs Cumulative Review

Chap 1-9Lesson 10-1

Exploring Quadratic GraphsNotesLesson 10-1

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


Identify the vertex of each graph and tell whether it is a minimum or a maximum.

Graphing y = ax2

(1)Make a table of values.

(2) 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 (-).


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 (-).

(2) 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.

Exploring Quadratic Graphs

HomeworkLesson 10-

1

Homework – Practice 10-1

#1-26

Quadratic Functions Practice 10-1Lesson 10-2

Quadratic Functions

Practice 10-1Lesson 10-2

Quadratic FunctionsNotesLesson 10-2

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 FunctionsNotesLesson 10-2

Graph y > x2 + x + 1

Quadratic Functions

HomeworkLesson 10-

2

Homework ~ Practice 10-2 even

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

Finding & Estimating Square Roots

NotesLesson 10-3

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.


NotesLesson 10-3

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


HomeworkLesson 10-

3

Homework – Practice 10-3 odd

Solving Quadratic Equations


Solving Quadratic EquationsNotesLesson 10-4

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 – 4 (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.

Solving Quadratic EquationsNotesLesson 10-4

t2 – 25 = 0

t2 = 25

t = ± 5

Find the solution(s) 3n2 + 12 = 12

Try… 2g2 + 32 = 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 = -3 x = 3 Solutions ±3

Solving Quadratic Equations

Homework

Lesson 10-4

Homework – Practice 10-4

odd

Factoring to Solve Quadratic Equations


~ Chapter 10 ~Chapter Review

Algebra I Algebra I

Chapter 10 Review Part 1

Chapter Review

Factoring to Solve Quadratic EquationsNotesLesson 10-5

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 = -7 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 = 12 or x = -4

Your turn… x2 + x – 12 = 0

x = -4 or x = 3

Factoring to Solve Quadratic EquationsNotesLesson 10-5

2x2 – 5x = 88

2x2 – 5x – 88 = 0

Your turn… Solve x2 – 12x = -36

x2 – 12x + 36 = 0

(x - 6)2 = 0

x = 6

Factoring to Solve Quadratic Equations

HomeworkLesson 10-5

~ Homework ~

Practice 10-5 even

Completing the Square


Using the Quadratic FormulaNotesLesson 10-7

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 ± 25 – 24 2

x = 5 ± √ 1 = 5 + 1 or 5 – 1 = 6 or 4 = 3 or 2 2 2 2 2 2

Using the Quadratic FormulaNotesLesson 10-7

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 ± 16 – (-56) 2(2) 4

x = -4 + √72 or -4 - √72 ≈ -4 + 8.49 or -4 - 8.49 ≈ 1.12 or -3.12 4 4 4 4

Your turn…

7x2 – 2x – 8 = 0

Using the Quadratic FormulaHomeworkLesson 10-7

Homework – Practice 10-7 even

Using the Discriminant



NotesLesson 10-8

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


Find the number of solutions for x2 – 2x – 3

b2 – 4ac = (-2)2 – 4(1)(-3)

4 – (-12) = 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


HomeworkLesson 10-8

Homework – Practice 10-8 odd

&

Chapter 10 Review Part 2


HomeworkLesson 10-8

~ Chapter 10 ~Chapter Review

Algebra I Algebra I

~ Chapter 10 ~ Quadratic Equations and Functions Algebra I Lesson 10-1 Exploring Quadratic Graphs...

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