Transcript Properties of Quadratics

Chapter 3
Properties of
Quadratics
Introduction of Quadratic
Relationships
The
graph of a quadratic is called a parabola.
The direction of the opening of the parabola
can be determined from the sign of the 2nd
difference in the table of values
the 2nd difference is positive then it opens
upwards.
If the 2nd difference is negative then it opens
downwards.
If
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Introduction of Quadratic
Relationships
The
vertex of a parabola is the point where
the graph changes direction. It will have the
greatest y-coordinate if it opens down or the
smallest y-coordinate if it opens up.
The y-coordinate of the vertex corresponds
to an optimal value. This can be either a
minimum or Maximum value
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Introduction of Quadratic
Relationships
A
parabola is symmetrical with respect to
vertical line through its vertex. This line is
called
the axis of symmetry.
If the coordinates of the vertex are (h, k),
then the equation of the axis of symmetry is
x = h.
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Introduction of Quadratic
Relationships
If
the parabola crosses the x-axis, the xcoordinates of these points are called the
zeros. The vertex is directly above or below
the midpoint of the segment joining the
zeros.
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Finding x-intercepts
Recall that in grade 9 math, we found the x-intercept of
linear equations by letting y = 0 and solving for x.
The same method works for x-intercepts in quadratic
equations.
Note: When the quadratic equation is written in standard
form, the graph is a parabola opening up (when a > 0) or
down (when a < 0), where a is the coefficient of the x2
term.
The intercepts will be where the parabola crosses the
x-axis.
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Finding x-intercepts
Example
Find the x-intercepts of the graph of y = 4x2 + 11x + 6.
The equation is already written in standard form, so
we let y = 0, then factor the quadratic in x.
0 = 4x2 + 11x + 6 = (4x + 3)(x + 2)
We set each factor equal to 0 and solve for x.
4x + 3 = 0 or x + 2 = 0
4x = –3 or x = –2
x = –¾ or x = –2
So the x-intercepts are the points (–¾, 0) and (–2, 0).
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