Home   
Real Number System   Absolute Value   Functions and Graphs   Trigonometry   Basic Graphing Skills   Studying a Function   Algebraic Skills   Answer Section 
Section 1  Section 2  Section 3   Section 4  Section 5   Section 6  Section 7  Section 8  Section 9  Section 10  Section 11  Section 12   Section 13  Section 14  Section 15  Section 16  Section 17  Section 18   Section 19   Section 20  Section 21  Section 22   Section 23   Answers 
Graphs of y1 = x^{2} and y2 = (x+1)^{2}  Table values of y1 = x^{2} and y2 = (x+1)^{2} 
Remember the base function y1 = x^{2} passes through (0,0), so y1 is the graph on the right. What happens to the graph when x is replaced with the quantity x+1? (The graph moves one unit to the left.)
Study the set of table values associated with each function to see how they are changed by replacing x with x+1.
(To obtain the same y values a value of x one less must be used. For example to obtain the value of 4 in y1 we use x = 2 or 2. But to obtain a value of 4 in y2 we need to use 1 or 3. These x values are one less.)
Graphs of y1=x and y2=3x  Table values of y1=x and y2=3x 
Remember the base function y1=x pass through the origin and the lines have slopes of 1 and 1. What happens to the graph when the function is multiplied by 3? (The values of the function are multiplied by 3 or they become larger.)
Study the set of table values associated with each function to see how they are changed by multiplying by 3.
(The y values from y1 are multiplied by 3 to get the y values for y2. This means the y values will be higher for each x value. So the graph of y=3x shows steeper lines. No shift will take place, just a vertical stretch.)
Graphs of y1=3^{x }and y2=3^{x}  Table values of y1=3^{x }and y2=3^{x} 
Remember the base function y1=3^{x }begins in quadrant II and passes through the (0,1) and continues through quadrant I. What happens to the graph when the x is replaced by x. (The graph is reflected over the yaxis.)
Study the set of table values associated with each function to see how they are changed by the replacement of x with x.
(Studying the y values you can notice that y2(1)=y1(1), y2(2)=y1(2) or in general the table for y1 has been turned upside down to produce the table for y2. This causes the graph to be reflected over the yaxis.)
Graphs of y1=3^{x }and y2=3^{x}  Table values of y1=3^{x }and y2=3^{x} 
Remember the base function y1=3^{x } ^{ }begins in quadrant III and passes through the point (0,1) and continues through quadrant I. What happens to the graph when the y=f(x) when it is changed to y=f(x)? (The graph is reflected over the xaxis.)
Study the set of table values associated with each function to see how the table values are changed by rewriting a function y=f(x) as y=f(x).
(The y values for y2 are opposite the y values for y1 for the same x values. The graph has been reflected over the xaxis.)
Graphs of y1=ln(x)^{ }and y2=ln(x)  Table values of y1=ln(x)^{ }and y2=ln(x) 
Remember the base function y1=ln(x)^{ } begins in quadrant IV and passes through the point (1,0) and continues through quadrant I. What happens to the graph when the y=f(x) when it is changed to y=f(x)? (There are two ways to look at the reflection: 1) The graph is reflected over the origin or 2) a double reflection has taken placefirst over the y axis and then over the x axis.)
Study the set of table values associated with each function to see how the table values are changed by rewriting a function y=f(x) as y=f(x).
(Studying the y values you can notice that y2(1)=y1(1), y2(2)=y1(2) or in general the table for y1 has been turned upside down to produce the table for y2 and the values have been changed to their opposite. This causes the graph to be reflected over the origin or the double reflection.)
Summary 

Vertical and horizontal shift transformations 

Original Function  Modified Function  Change 
If y = f(x) 
y = f(x)+ a  shifts the function vertically a units 
y = f(x)  y = f(xa)  shifts the function horizontally a units. 
Stretching and shrinking transformations  
Original Function  Modified Function  Change 
y = f(x)  y = af(x)  stretches the function vertically if a>1 
y = f(x) 
y = af(x)  shrinks the function vertically if 0<a<1 
Reflections  
Original Function  Modified Function  Change 
y = f(x) 
y = f(x)  reflect the function about the yaxis 
y = f(x)  y = f(x)  reflect the function about the xaxis 
y = f(x)  y = f(x)  reflect the function about the origin 

Note , so each point (x,y) on the original function is now transformed to (x, 3x+6). 
Vertical shift: 6Stretch: 3No reflection 

Note , so each point (x,y) on the original function is now transformed to (x, 6x+2). 
Vertical shift: 2Stretch: 6Reflection: about the xaxis 

Horizontal shift: 2No vertical shiftVertex: (0,2)No stretching or shrinkingNo reflection 

No horizontal or vertical shiftStretching factor: 3 (steeper) 

In this new function the x has be interchanged with y to create an equation which is no longer a function of x but instead a function of y. Note the change in the function. 

Horizontal shift: 3Vertical Shift: 3Vertex: (3,3)Shrinking factor: 0.1Curvature: less steep 

Center: (0,0)Radius: 4 
(The reason the whole circle does not appear is because there is not pixel representing the point (1,4) or (9,4). If a different window had been selected, the circle might appear complete.) 
Vertical shift: 4Horizontal shift: 5Center: (5,4)Radius: unchanged 

Vertical shift: 2Horizontal shift: 3Vertex: (3,2)Radius: 5 (increased by 1) 

Vertex: (0,0)Equivalent piecewise equations: 

Note B(x)=A(3x)No horizontal or vertical shiftStretching factor: 3Two steeper linesPiecewise equations: 

Horizontal shift: 2Vertex: (2,0)Piecewise Equations:No stretching or shrinking 
Horizontal shift: 1/2Vertical Shift: 2Vertex: (1,2)Stretching factor: 2Two steeper linesPiecewise equations: 
Describe what two
transformations have been performed on the base function y1 to produce
the new set of table values for y2.
This page was modified on 03/21/16
© Rahn, 2000