Section 5.1 Introduction to Quadratic Functions. Quadratic Function A quadratic function is any function that can be written in the form f(x) = ax² +

Section 5.1

Introduction to Quadratic Functions

Quadratic Function

• A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a ≠ 0.

• It is defined by a quadratic expression, which is an expression of the form as seen above.

• The stopping-distance function, given by: d(x) = ⅟₁₉x² + ¹ ₁₀x, is an example of a quadratic ¹̸�function.

Quadratic Functions

• Let f(x) = (2x – 1)(3x + 5). Show that f represents a quadratic function. Identify a, b, and c.

• f(x) = (2x – 1)(3x + 5)• f(x) = (2x – 1)3x + (2x – 1)5• f(x) = 6x² - 3x + 10x – 5 • f(x) = 6x² + 7x – 5 a = 6, b = 7, c = - 5

Parabola

• The graph of a quadratic function is called a parabola. Parabolas have an axis of symmetry, a line that divides the parabola into two parts that are mirror images of each other.

• The vertex of a parabola is either the lowest point on the graph or the highest point on the graph.

Domain and Range of Quadratic Functions

• The domain of any quadratic function is the set of all real numbers.

• The range is either the set of all real numbers greater than or equal to the minimum value of the function (when the graph opens up).

• The range is either the set of all real numbers less than or equal to the maximum value of the function (when the graph opens down).

Minimum and Maximum Values

• Let f(x) = ax² + bx + c, where a ≠ 0. The graph of f is a parabola.

• If a > 0, the parabola opens up and the vertex is the lowest point. The y-coordinate of the vertex is the minimum value of f.

• If a < 0, the parabola opens down and the vertex is the highest point. The y-coordinate of the vertex is the maximum value of f.

Minimum and Maximum Values

• f(x) = x² + x – 6 • Because a > 0, the

parabola opens up and the function has a minimum value at the vertex.

• g(x) = 5 + 4x - x²• Because a < 0, the

parabola opens down and the function has a maximum value at the vertex.

Section 5.2

Introduction to Solving Quadratic Equations

Solving Equations of the Form x² = a

• If x² = a and a ≥ 0, then x = √a or x = - √a, or simply x = ± √a.

• The positive square root of a, √a is called the principal square root of a.

• Simplify the radical for the exact answer.

Solving Equations of the Form x² = a

• Solve 4x² + 13 = 253

• 4x² + 13 = 253 Simply the Radical - 13 - 13 √60 = √(2 2 ∙ 3 5)∙ ∙ 4x² = 240 √60 = 2√(3 5)∙

√60 = ± 2√15 (exact answer)4x² = 240

4 4 x² = 60 x = √60 or x = - √60 (exact answer) x = 7.75 or x = - 7.75 (approximate answer)

Properties of Square Roots

• Product Property of Square Roots:• If a ≥ 0 and b ≥ 0: √(ab) = √a √∙ b

• Quotient Property of Square Roots:• If a ≥ 0 and b > 0: √(a/b) = √(a) ÷ √(b)

Properties of Square Roots

• Solve 9(x – 2)² = 121

• 9(x – 2)² = 121 x = 2 + √(121/9) or 2 - √(121/9) 9 9

x = 2 + [√(121) / √ (9)] or 2 – [√(121) / √(9)]

(x – 2)² = 121/9 x = 2 + (11/3) or 2 – (11/3) √(x – 2)² = ±√(121/9) x = 17/3 or x = - 5/3 x – 2 = ±√(121/9)

x – 2 = √(121/9) + 2 + 2

Pythagorean Theorem

• If ∆ABC is a right triangle with the right angle at C, then a² + b² = c²

A

c a

C B b

Pythagorean Theorem

• If ∆ABC is a right triangle with the right angle at C, then a² + b² = c²

A2.5² + 5.1² = c²

c 6.25 + 26.01 = c² 2.5 32.26 = c²

√(32.26) = c C B 5.68 = c 5.1

Section 5.3

Factoring Quadratic Expressions

Factoring Quadratic Expressions

• When you learned to multiply two expressions like 2x and x + 3, you learned how to write a product as a sum.

• Factoring reverses the process, allowing you to write a sum as a product.

• To factor an expression containing two or more terms, factor out the greatest common factor (GCF) of the two expressions.

Factoring Quadratic Expressions

• 3a² - 12a 3x(4x + 5) – 5(4x + 5)

• 3a² = 3a a∙ The GCF = 4x + 5• 12a = 3a 4∙ (3x – 5)(4x +5)• The GCF = 3a• 3a(a) – 3a 4∙• (3a)(a – 4)

Factoring x² + bx + c

• To factor an expression of the form:• ax² + bx + c where a = 1, look for integers r and

s such that r s = c ∙ and r + s = b. • Then factor the expression.• x² + bx + c = (x + r)(x + s)

Factoring x² + bx + c

• x² + 7x + 10 x² - 7x + 10

(5+2) = 7 & (5 2) = 10∙ (-5-2) = -7 & (-5 (-2)) = ∙10

(x + 5)(x + 2) (x – 5)(x – 2)

Factoring the Difference of Two Squares

• a² - b² = (a + b)(a – b)• (x + 3)(x – 3)• x² + 3x - 3x - 9• x² - 9• x² - 3²

Factoring Perfect-Square Trinomials

a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a – b)²(x + 3)² (x – 3)²(x + 3)(x + 3) (x – 3)(x – 3)x² + 3x + 3x + 9 x² - 3x – 3x + 9X² +2(3x) + 9 x² - 2(3x) + 9

Zero-Product Property

• If pq = 0, then p = 0 or q = 0.

• 2x² - 11x = 0• x(2x – 11) = 0 (Factor out an x)• x = 0 or 2x – 11 = 0• x = 11 or x = 11/2

Section 5.4

Completing the Square

Completing the Square

• When a quadratic equation does not contain a perfect square, you an create a perfect square in the equation by completing the square.

• Completing the square is a process by which you can force a quadratic expression to factor.

Specific Case of a Perfect-Square Trinomial

• x² + 8x + 16 = (x + 4)²

• Understand: (½)8 = 4 → 4² = 16

Examples of Completing the Square

• x² - 6x x² + 15x• (½)(-6) = -3 (½)(15) = (15/2)• (-3)² → = 9 (15/2)² → = (15/2)²• The perfect-square The perfect square• x² - 6x + 9 = x² + 15x + (15/2)² =• (x – 3)² [x + (15/2)]²

Solving a Quadratic Equation by Completing the Square

• x² + 10x – 24 = 0 2x² + 6x = 7• + 24 +24 2(x² + 3x) = 7• x² +10x = 24 x² + 3x = (7/2)• x²+10x+(5)²=24+(5)² x²+3x+(3/2)²=(7/2)+(3/2)²• x² + 10x + 25 = 49 x²+3x+(3/2)² =(7/2)+(9/4)• (x + 5)² = 49 [x + (3/2)]² = (23/4)• x + 5 = ± 7 x + (3/2) = ±√(23/4)• x = - 12 or x = 2 x = - (3/2) + √(23/4) (0.90)

X = - (3/2) - √(23/4) (-3.90)

Vertex Form

• If the coordinates of the vertex of the graph of y = ax² + bx + c, where a ≠ 0, are (h,k), then you can represent the parabola as:

• y = a(x – h)² + k, which is the vertex form of a quadratic function.

Vertex Form

• Given g(x) = 2x² + 12x + 13• 2(x² + 6x) + 13• 2(x² + 6x + 9) + 13 – 2(9)• 2(x + 3)² - 5• 2[x – (-3)]² + (- 5) (Vertex Form)• The coordinates (h,k) of the vertex are (-3, -5)

and the equation for the axis of symmetry is x = - 3.