Graphing Quadratic Functions Worksheet
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4Graphing quadratic functions.[rev.4] Class/Section _______Name __________________ Date ________
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y = ax
+bx+c
A quadratic function is a function of the form
, where a, b, and c are real
numbers.
Skill #1. Evaluating a quadratic function.
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Example. f(x) = 3x
-4x-5. Evaluate f(x) for x= -2/3.
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Answer: f(-2/3) = 3(-2/3)
-4(-2/3)-5 = 3(4/9) -4(-2/3) -5 = 4/3 + 8/3 - 15/3 = -3/3 = -1.
Skill #2. Calculating the discriminant.
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A quadratic function has a discriminant Δ = b
-4ac.
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The discriminant of the function f(x) = 3x
-4x-5 is calculated as follows.
First write down a,b, and c:
Then write:
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Δ = b
-4ac
a = 3
2
– 4 ( ) ( )
= ( )
b = -4
2
– 4 (3 ) (- 5)
= ( -4)
c = -5
= 16 + 60 = 76.
Skill #3. Using the discriminant.
The discriminant Δ tells you several things.
(a) If Δ is a perfect square integer (Δ = 0,1,4,9,16,25, etc), then the quadratic
polynomial factors over the integers.
(b) If Δ > 0, then the quadratic function has two different real roots.
(c) If Δ = 0, then the quadratic function has one real double root.
(d) If Δ < 0, then the quadratic function has no real roots, but it has two complex
conjugate roots.
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Example: the polynomial function f(x) = 3x
-4x-5 has discriminant 76. Since 76 is not a
perfect square, the polynomial cannot be factored over the integers. Since 76 > 0, the
polynomial has two different real roots.
Skill #4. Does the parabola open up or down?
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∪
The graph of a quadratic function opens upwards (
) if the coefficient of x
is a positive
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number. The graph opens downwards (∩) if the coefficient of x
is a negative number.
Skill #5. Finding the x-coordinate of the vertex, and the line of symmetry (axis of symmetry).
The vertex of a parabola (or quadratic function) is the (x,y) point which is the highest or
lowest point on the curve. The x-value of the vertex is half-way between the two roots,
or the average of the two roots. It may be calculated directly by the formula x = -b/(2a) .
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Example: the x-value of the vertex of the equation f(x) = 3x
-4x-5 is
x= - b/(2a) = - (-4)/(2(3)) = 4/6 = 2/3.
The line of symmetry (axis of symmetry) of the parabola is the vertical line through
the vertex.
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