– Unit 6: Absolute Value & Quadratic Models

A 606. Solve absolute value equations (Ch. 7.5) N 404. Understand absolute value in terms of distance (Ch. 7.5 & 11.6 ) A 509. Work with squares and square roots of numbers (Ch. 7.6)

N 031. Recognize one-digit factors of a number A 404. Multiply two binomials A 506. Identify solutions to simple quadratic equations A 507. Solve quadratic equations in the form (x + a)(x + b) = 0 A 508. Factor simple quadratics (e.g., the difference of squares & perfect square trinomials) A 605. Solve quadratic equations

Ch. 7.5 Defining the Absolute-Value Function Learning Intentions:

Investigate the concept of absolute value.

Construct & interpret graphs of absolute-value functions.

Evaluate expressions containing absolute value.

Absolute Value: is the distance between a number’s corresponding point on a number line & the origin, 0.

Because it is a distance, an absolute value is NEVER negative.

Ex.) Simplify.

5 = _____ −5 = _____ −1.5 = ____ 1.5 = _____

-5 -1.5 0 1.5 5 Ex.) Evaluate the following expression.

1.) −4.5 2.) 1 – −7 3.) 1 − 7

_______ _______ _______

Notation: ABS(x) or 𝒙

Because it is a distance, an absolute value is NEVER negative. Ex.) Simplify.

5 = 5 −5 = 5 −1.5 = 1.5 1.5 = 1.5

-5 -1.5 0 1.5 5 Ex.) Evaluate the following expression.

1.) −4.5 2.) 1 – −7 3.) 1 − 7

1 – 7 −6

4.5 -6 6

Absolute Value: is the distance between a number’s corresponding point on a number line & the origin, 0. Notation: ABS(x) or 𝒙 SOLUTIONS:

y = 𝒙 y = – 𝒙 y = 𝒙 + 4 y = 𝒙 + 𝟒 x y x y x y x y -3

-2

-1

0

1

2

3

See p.420-421 for additional examples.

How do the additional operations affect the graphs when compared to the parent function, y = 𝒙 ? Graph each absolute-value function.

y = 𝒙 y = – 𝒙 y = 𝒙 + 4 y = 𝒙 + 𝟒

x y x y x y x y

-3 3 -3 -3 -3 7 -6 2

-2 2 -2 -2 -2 6 -5 1 -1 1 -1 -1 -1 5 -4 0

0 0 0 0 0 4 -3 1

1 1 1 -1 1 5 -2 2

2 2 2 -2 2 6 -1 3

3 3 3 -3 3 7 0 4

SOLUTIONS: Graph each absolute-value function.

How do the additional operations affect the graphs when compared to the parent function, y = 𝒙 ?

NOTE:

Solving Equations Involving Absolute Value: Ex.) Solve for x.

1.) 𝒙 = 10 2.) 𝒙 − 𝟒 = 6 3.) 𝒙 = -5 x = ___ & x = ___

4.) 𝒙 = 12 5.) 10 = 𝒙 + 4 6.) 10 = 2 𝒙 + 6

Solving Equations Involving Absolute Value: Ex.) Solve for x.

1.) 𝒙 = 10 2.) 𝒙 − 𝟒 = 6 3.) 𝒙 = -5 x – 4 = 6 or x – 4 = -6 x = no solution x = 10 or x = -10 x = 10 or x = -2

4.) 𝒙 = 12 5.) 10 = 𝒙 + 4 6.) 10 = 2 𝒙 + 6 x = 12 or x = -12 -4 -4 -6 -6

6 = 𝒙 4 = 2 𝒙 2 = 𝒙 x = 6 or x = -6 x = 2 or x = -2

If possible, solve each equation for x. Check your answer by substituting the value or values into the original equation.

7.) 𝑥 + 1 = 7 8.) 2 3𝑥 + 1 = 4

SOLUTIONS: If possible, solve each equation for x. Check your answer by substituting the value or values into the original equation.

7.) 𝑥 + 1 = 7 8.) 2 3𝑥 + 1 = 4 |3x + 1| = 2 x + 1 = 7 or x + 1 = -7

x = 6 or x = -8 3x + 1 = 2 or 3x + 1 = -2

x = 𝟏

𝟑 or x = -1

Check: Let x = 6 Let x = -8 𝑥 + 1 = 7 𝑥 + 1 = 7 𝟔 + 1 = 7 −𝟖 + 1 = 7 𝟕 = 7 −𝟕 = 7

7 = 7 7 = 7

Check:

Let x = 𝟏

𝟑 Let x = -1

2 3𝑥 + 1 = 4 2 3𝑥 + 1 = 4

2 3(𝟏

𝟑) + 1 = 4 2 3(-1) + 1 = 4

2 𝟐 = 4 2 −𝟐 = 4 2(2) = 4 2(2) = 4 4 = 4 4 = 4