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Functions and Their Graphs
5 basic graphs
Formula:
1) y = x
2) y = x²
3) y = x³
4) y = √x
5) y = |x|
Graph:
1) Linear— “line”
2) Quadratic— “parabola”
3) Cubic— “squiggly”
4) Square Root– “half of a parabola
5) Absolute Value– “V-shaped”
Shiftsy = a(x - h)²+k
h is the horizontal shift. The graph will move in the opposite direction.
K is the vertical shift. The graph will move in the same direction.
If a is positive, then the graph will go up. If a is negative, then the graph will go down.
If a is positive, then the graph will go up.
If a is negative, then the graph will go down.
More info about parabolasy = a(x – h)² + k
The Vertex of a Parabolay = a(x – h)² + k
The vertex is (h, k). In other words, the vertex is (H.S., V.S). For example, y = (x – 2)² – 9. H.S. = Right 2 V.S. = Down 9 Therefore, the vertex is denoted by V(2, -9).
For example: y = x² + 6
y = (x – 0)² + 6, so the V.S. is up 6 and the H.S. is none. Therefore, the vertex is
V (0, 6). Since a is positive, the direction of the
parabola is up. Since a is 1, then the parabola is neither fat
or skinny. It is a standard parabola.
Another example: y = 2(x + 2)² + 6
H.S. = left 2 V.S. = up 6 V (–2, 6) Direction is up Since a = 2, then the parabola is skinny
Axis of Symmetry of a Parabolay = a(x – h)² + k
x = H.S. or x = h
For example, y = (x – 2)²– Axis of symmetry is x = 2
Practice
up or down? fat or skinny? V.S.? H.S.? Axis Symmetry: Graph it
21f(x)= x+4 -52
Practice 21f(x)= x+4 -52
Practicey = -|x – 6| + 3
What does this graph look like? Horizontal shift? Vertical shift? Vertex? Fat or skinny? Up or down?
Practicey = -|x – 6| + 3
Practicey = (x + 1)³ - 5
What does this graph look like? H.S.? V.S.? There is no vertex. Right or Left? Fat or skinny?
Practicey = (x + 1)³ - 5
Practice
What does this graph look like? H.S.? V.S.? There is no vertex. Up or down? Fat or skinny?
2 5y x
Practice 2 5y x
Domain and Range
Domain is the set of all x-values. You will look at the graph from left to right (like you’re reading a book). Ask yourself: where does the graph begin? Where does it end?
Range is the set of all y-values. You will look at the graph from bottom to top. Ask yourself: where does the graph begin? Where does it end?
Find the domain and range.
From left to right, while following the x-axis, where does the graph begin? Where does it end?
From bottom to top, while following the y-axis, where does the graph begin? Where does it end?
Answer:
Symmetry
Symmetry
Symmetry
Symmetry
Any questions?
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