Functions and Their Graphs Picture is from Microsoft Clip Art

5 basic graphs Formula: y = x y = x² y = x³ y = √x y = |x| Graph: Linear— “line” Quadratic— “parabola” Cubic— “squiggly” Square Root– “half of a parabola Absolute Value– “V-shaped”

Shifts y = 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 positive, then the graph will go up. If a is negative, then the graph will go down.

More info about parabolas y = a(x – h)² + k    

The Vertex of a Parabola y = 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 Parabola y = 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

Practice  

Practice y = -|x – 6| + 3 What does this graph look like? Horizontal shift? Vertical shift? Vertex? Fat or skinny? Up or down?

Practice y = -|x – 6| + 3  

Practice y = (x + 1)³ - 5 What does this graph look like? H.S.? V.S.? There is no vertex. Right or Left? Fat or skinny?

Practice y = (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?

Practice  

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?