Section 1.1 Parent Functions and Transformations Honors Algebra II Section 1.1 Parent Functions and Transformations

Essential Question What is a relation?

Relation- set of ordered pairs each x does not necessarily have a unique y

Essential Question What is a function?

Function-a relation in which every input (x) is paired with exactly one output (y or f(x)) *each x has a unique y *y may be paired with more than one x

Domain- set of all possible input values in a relation or function (independent variable) Range- set of all possible output values in a relation or function (dependent variable)

https://www.youtube.com/watch?v=VUTXsPFx-qQ

Essential question What are the parent functions?

Parent Functions #1 Constant Function y= constant f(x) = constant Graph is a horizontal line Domain- all real numbers Range- the constant in the equation

#2 Linear Function y=x f(x) = x Graph is a straight line #2 Linear Function y=x f(x) = x Graph is a straight line. Domain- all real numbers Range- all real numbers

#3 Quadratic function y= 𝑥 2 f(x) = 𝑥 2 Graph is a parabola #3 Quadratic function y= 𝑥 2 f(x) = 𝑥 2 Graph is a parabola. Domain- all real numbers Range- {y|y ≥ 0}

#4 Cubic function y= 𝑥 3 f(x) = 𝑥 3 Graph looks like the union of the right side of an up parabola and the left side of a down parabola Domain- all real numbers Range-all real numbers

#5 Square root function y = 𝑥 f(x) = 𝑥 Graph looks like the top half of a parabola turned to the right. Domain= {x|x≥ 0} Range= {y|y≥ 0}

#6 Absolute Value Function 𝑦= 𝑥 𝑓 𝑥 = 𝑥 Graph looks like a “v” Domain: all real numbers Range 𝑦 𝑦≥0

#7 Exponential function 𝑦= 𝑎 𝑥 𝑓 𝑥 = 𝑎 𝑥 The graph keeps getting steeper Domain: all real numbers Range: 𝑦 𝑦>0

#8 Reciprocal function 𝑦= 1 𝑥 𝑓 𝑥 = 1 𝑥 Graph looks like two curves approaching the x and y axes. Domain: {All reals except x=0} Range: {All reals except y=0}

Essential Question What is a transformation?

Change can mean: translation- moving (left/right, up/down) reflection- flipping stretching/shrinking

Graphing parent functions and transformations Step 1 Set up graph paper on the calculator The most common window is shown below. Setting up a window is like drawing a grid on graph paper.

Step 2 Type the equation into y= Press graph

Typing in the equation There is an x key (next to the alpha key) If you need a negative sign (it is on the bottom row). Do not use the subtraction sign! There is a squared key. For any exponent bigger than two use the carrot key (below clear)

Finding some points on the graph Press the second key Press the graph key (table) It is usually set up so the x values increase by 1. This can be changed.

You can graph more than one equation on the calculator Type 1st equation into 𝑦 1 Type 2nd equation into 𝑦 2

Translations What happens when you add a number to the x for the function 𝑦= 𝑥 2 ? Use parentheses around the quantity squared. Try this on the calculator! 𝑦 1 = 𝑥 2 𝑦 2 = (𝑥+4) 2 What would happen for 𝑦 3 = 𝑥 2 +5?

What changes occurred in these graphs? What parent graph is in black?

Moving left or right affects the _________ coordinate Moving up or down affects the

Add to “x”, go left Add to “y”, go high

Reflections What happens when you put a negative into one of the parent functions? Different things happen when the negative is inside versus outside parentheses. Consider 𝑦= 𝑥 3 𝑦=− 𝑥 3 𝑦=(− 𝑥 3 )

Minimum number of points to make a good graph Type of graph Number of points Constant Linear Quadratic Cubic Square root Absolute Value Exponential Reciprocal

When a graph is a transformed, the domain and range may change.

Assignment #1 Pg. 8 #1-45 (do 27-33 without a graphing calculator)