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?