a Use the properties of transformations to determined the type of transformation. Start by graphing the parent function, use the vertex and two additional points. Then, use those same points to move the parent function 3 units up.
B
b Use the parent function from Part A to double the function values and graph the vertical expansion.
C
c Since the effect of this transformation is on the x-values, use the points parent function from Part A to divide each x-coordinate by 4 to get the new points for the transformation.
D
d Since the effect of this transformation is on the x-values, use the parent function from Part A to subtract 5 from each x-coordinate to get the new points for the transformation.
A
aTransformation: Vertical translation 3 units up Graph:
B
bTransformation: Vertical stretch by a factor of 2 Graph:
C
cTransformation: Horizontal compression by a factor of 4 Graph:
D
dTransformation: Horizontal shift 5 units left Graph:
Practice makes perfect
a We will start this part of the exercise by plotting points for the parent function then use those points to graph the transformation.
Parent Function
We will use a 3 step approach to graphing the parent function.
Find the axis of symmetry and vertex
Find two other points
Graph
Vertex
Our parent function, f(x)=x^2+2x-3, is already in standard form, y= ax^2+ bx+ c, so we can go directly to finding the axis of symmetry.
x=- b/2 a
In our function, f(x)=x^2+2x-3, we can substitute a= 1, b= 2, and c= 3 into the formula for the axis of symmetry.
From this, we can conclude that our vertex is (-1,-4).
Other Points
Three non-linear points define a parabola, so we only need two additional points to graph this function. Let's make a table of values.
x
f(x)=x^2+2x-3
y=f(x)
0
0^2+2( 0)-3
-3
2
2^2+2( 2)-3
5
Now we have the additional points (0,-3) and (2,5). Let's look at the graph.
Plotting Points
We can start this graph by plotting the three points that we have, adding in the axis of symmetry, adding in symmetric points, then connecting the points with the parabola.
Transformation f(x)+3
Let's first look at what kind of transformation we have.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Our function, f(x)+ 3, looks most like an vertical translation upward. We can see that the + 3 will shift the function up 3 units. Let's add 3 to each of the y-coordinates of our three points from our parent function and see what this transformation does to them.
(x,f(x))
(x,f(4x))
(-1,-4)
(-1,-1)
(0,-3)
(0,0)
(2,5)
(2,8)
Let's plot these points with our parent function, then move the mirrored points up as well.
b We can graph our parent function like in Part A, then plot the transformed points below. Let's first look at what kind of transformation we have.
Transformations of f(x)
Vertical Stretch or Compression
Vertical stretch, a>1
y= af(x)
Vertical compression, 0< a< 1
y= af(x)
Our function, 2[f(x)], looks most like a vertical stretch. The 2[f(x)] stretches the function up and down by a factor of 2. Let's have a look at our three points from our original function and multiply the y-coordinates by 2.
(x,f(x))
(x,2[f(x)])
(-1,-4)
( -1,-8)
(0,-3)
( 0,-6)
(2,5)
( 2,10)
Let's plot those points and the mirrored points from the parent function.
c We will plot the parent function from Part A, and, then, the transformation. Let's first look at what kind of transformation we have.
Transformations of f(x)
Horizontal Stretch or Compression
Horizontal stretch, 0< b<1
y=f( bx)
Horizontal compression, b>1
y=f( bx)
Our function, f( 4x), is a horizontal compression and shrinks the function by a factor of 4. When a transformation is inside the parentheses with the x, then the transformation affects the x-axis in the opposite way of the operation on x. Let's divide each x-coordinate by our three points from our parent function by 4.
(x,f(x))
(x/4,f(x))
(-1,-4)
( -1/4,-4)
(0,-3)
( 0,-3)
(2,5)
( 1/2,5)
Let's plot those points and the mirrored points from the original function.
d For this part of the exercise, we will graph the parent function, and, then, plot the corresponding points for the transformation. Let's first look at what kind of transformation we have.
Transformations of f(x)
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
Our function, f(x+ 5), is a horizontal shift that shifts the function to the left by 5 units. When a transformation is inside the parentheses with the x, then the transformation affects the x-axis in the opposite way of the operation on x. Let's subtract 5 from each x-coordinate of the three points from our parent function in Part A.
(x,f(x))
(x-5,f(x))
(-1,-4)
( -6,-4)
(0,-3)
( -5,-3)
(2,5)
( -3,5)
Let's plot those points and the mirrored points from the original function.
