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Graph of quadratic functions
We start with a simple graph of y = x2.
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y = x2
x
y
Vertex(0, 0)
Important features
It is shaped.
It is symmetrical about a line x = 0 (i.e. y axis).
It has a vertex at (0,0) (i.e. the minimum point).
Graph of quadratic functions
By changing the equation slightly, we can shift the curve around without changing the basic shape.
y = x2 + 5
x
y
The graph of y = x2 + 5 can be obtained by translating the graph of y = x2 five units in the y-direction.
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Vertex (0, 5)
Graph of quadratic functions
The graph of y = x2 – 10 can be obtained by translating the graph of y = x2 ten units in the negative y direction.
x
y
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y = x2 - 10
Vertex (0, -10)
Graph of quadratic functions
I we replace x by x – k in the equation of a graph then the graph produces a translation of k units in the x direction.
x
y
The graph of y = (x – 2)2 can be obtained by translating the graph of y = x2 two units in the x direction.
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y = (x – 2)2
Vertex (2, 0)
Graph of quadratic functions
In a similar fashion, the graph of y = (x + 4)2 is a shift
of – 4 in the x-direction, the vertex is at (-4, 0).
x
y
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y = (x + 4)2
Vertex (-4, 0)
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Graph of quadratic functionsWe start with a simple graph of y = -x2
Important features
It is shaped.
It is symmetrical about a line x = 0 (i.e. y axis).
It has a vertex at (0,0) (i.e. the maximum point).
y = - x2
Vertex(0, 0)
y
x
Graph of quadratic functions
We can also have combinations of these transformations: The graph of y = (x – 2)2 – 10 has a shift of 2 units in the x-direction and –10 in the y-direction, with minimum point at (2, -10).
x
y
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y = (x – 2)2 - 10
Vertex (2, -10)
Use of the discriminant b2 – 4ac The discriminat of the quadratic function y = ax2 + bx + c is the
value of b2 – 4ac.
Discriminat
b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0
Number of roots: two one None
Intersection with the x-axis Two points Touch at one point Do not meet
Sketch a >0
Sketch a < 0
