What You’ll Learn: Identifying families of functions for equations and graphs. To predict what the graph of the equation looks like. What You’ll Need: Graphing calculator Note cards

Work with a partner. 1.Graph each equation using the standard range setting. To help you keep track, sketch each graph on a note card. Label it with the correct equation. y = x 2 – 6y = x + 2 y = | x | – 4y = 7x y = | x – 3 | y = x y = 3x 2 y = -3x

2.A.Sort the cards into three categories by grouping graphs that look alike. B.What similarities among the graphs in each category do you see? C.What similarities among the equations in each category do you see? The categories you made can help you make predictions.

3.What does the graph of y = 2x 2 look like? 4.What can you say about the equation of this graph?

You’ve already seen how grouping functions that are alike can help you make predictions. These groups are called families of functions. You can identify what family a function belongs to by looking at its equation.

To what family of functions does each equation belong? Explain. A.y = 2x – 6B.y = -8x 2 Its highest power of x is 1. Its highest power of x is 2. So, y = 2x – 6 is a So, y = -8x 2 is a linear function. quadratic function.

5.The equation y = | x + 7 | is an absolute value function. What characteristic of the equation tell you this?

6.To what family of functions does each equation belong? EXPLAIN. A.y = 6x B.y = 3 | x | C.y = x 2 + 3x + 2

7.Create three equations that belong to the quadratic family of functions. y = 4x 2 – 2 y = -x 2 y = 2x 2 – x + 1

You can identify what a family of function belongs to by looking at its graph. To what family of functions does each graph belong? Explain. a.b.

8.A.The equation y = -5x belongs to what family of functions? How do you know? B.Look at the graph of y = -5x. What characteristic of a graph tells you it belongs to the linear family of functions? Linear; the highest power of x is 1. The graph is a straight line.

y = x + 2 The highest power of x is 1. y = -x The highest power of x is 2. y = |x+2| There is an absolute value symbol around a variable expression..

To what family of functions does each equation belong? Explain. 1. _________________________ 2. _________________________ 3. _________________________ 4. _________________________ Linear function; the highest power of x is 1. Quadratic function; the highest power of x is 2. Quadratic function; the highest power of x is 2. Quadratic function; the highest power of x is 2.

To what family of functions does each equation belong? Explain. 5. _________________________ 6. _________________________ 7. _________________________ 8. _________________________ Quadratic function; the highest power of x is 2. Absolute value function; There is a variable expression inside the absolute value symbol. Absolute value function; There is a variable expression inside the absolute value symbol. Linear function; the highest power of x is 1.

To what family of functions does each equation belong? Explain your reasoning. Absolute Value; graph forms a “v” that opens up. Absolute Value; graph forms a “v” that opens down. Quadratic; graph is a U- shaped curve that opens down.

12.The recommended dosage D in milligrams of a certain medicine depends on a person’s body weight w in kilograms. To what family of functions does the formula D = 0.1w 2 + 5n belong? Explain. Quadratic; the highest power of x is 2.

13.Why are these graphs not quadratic or absolute value functions? They fail the vertical- line test. 14.Is a vertical line the graph of a linear function? Why or why not? No; it fails the vertical-line test so it is not a function. 15.Write two linear, two quadratics, and two absolute value equations. Samples: y = x + 1y = 3x + 6 y = x 2 + xy = -3x y =  x + 2  y = -  x + 4  - 1

What characteristics do you look for to identify the families of each? Function Characteristic Graph of a quadratic function Equation of a linear function Graph of an absolute value function Equation of a quadratic function

Determine to which family of functions each graph belongs. Then sketch a model each situation. 20.Income is a function of hours worked. 21.Height of a fly ball is a function of time. linear quadratic