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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 + ¹¹̸₁₀x, is an example of a quadratic
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
- 13 - 13
4x²
= 240
Simply the Radical
√60 = √(2 ∙ 2 ∙ 3 ∙ 5)
√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
9
9
(x – 2)² = 121/9
√(x – 2)² = ±√(121/9)
x – 2 = ±√(121/9)
x–2
+2
= √(121/9)
+2
x = 2 + √(121/9) or 2 - √(121/9)
x = 2 + [√(121) / √ (9)] or 2 – [√(121) / √(9)]
x = 2 + (11/3) or 2 – (11/3)
x = 17/3 or x = - 5/3
Pythagorean Theorem
• If ∆ABC is a right triangle with the right angle
at C, then a² + b² = c²
A
c
a
C
B
b