# 1.3 Notes - Transformations of Function Graphs (1)

```1-3 Transformations of Function Graphs
Function graphs can be transformed by translating them left, right, up, and down. They can be stretched or
compressed vertically or horizontally, and they can be reflected across the x-axis or the y-axis. The general
format that we will use for transformed functions looks like this:
 1

f ( x) = ±a ⋅ f  ± ( x − h)  + k
 b

Translations
When a value k is added to a function, the function’s graph is translated vertically. When a value h is
subtracted from x within the function, the graph is translated horizontally.
Vertical Example:
Consider this generic function f(x):
If we add a k-value of 4 to this function,
we change it by adding 4 to each y-value.
If we add a k-value of -3 to this function,
we subtract 3 from each y-value.
The translated graphs would look like this:
When k = 4, the graph is shifted up 4 units.
When k = -3, the graph is shifted down 3 units.
Horizontal Example:
Using the same generic function f(x) as above, let h = 2 so that we are now graphing f(x-2)
Each x-value that we choose has the y-value of what used to be 2 units before it.
An easier way of saying this is the graph is shifted 2 units to the right.
**Remember, the horizontal shift is the opposite of the sign that you see since we
are subtracting h from the x-value.
If h is a negative number, then f(x- -h) is f(x + h) and it shifts to the left.
Stretches and Compressions
If you multiply by a parameter outside the function, it will stretch or compress the graph vertically. If you
multiply by a parameter inside the function, it will stretch or compress the graph horizontally.
Vertical Example:
Taking the same function from the previous examples, let a = 2 so that we have g(x) = 2f(x)
Because each y-value is multiplied by 2, the
graph is now stretched vertically by a factor
of 2.
If we let a = ½, each y-value gets multiplied by ½ and the graph is compressed vertically by a factor of 2.
Horizontal Example:
Now, let’s let b = 2. This means we have g(x) = f(½x).
Now each x that we choose gets the y-value of what half of that xvalue had in the original function. An easier way of saying this is
the graph is stretched horizontally by a factor of 2.
**Remember, the horizontal stretch is the reciprocal of the value
you see since we are multiplying by 1/b.
If b is between 0 and 1, you will have a horizontal compression. And 1/b will be a number greater than 1.
1
b


1
b


**It is important to note the difference between g ( x) = f  ( x − h)  and g ( x) = f  x − h  . The 1/b MUST
BE FACTORED OUT in order to get the correct value for the horizontal shift.
Example: g ( x) = f ( 2x − 8) appears to be the function compressed by a factor of 2 and shifted right 8.
However, it is actually only shifted right 4 because in the proper form it would be g ( x) = f ( 2( x − 4) ) .
Reflections
If a is negative, the graph will reflect across the x-axis. If b is negative, the graph will reflect across the y-axis.
y = -f(x)
y = f(-x)
Modeling with a Quadratic Function
The netting of an empty hammock hangs between its supports along a curve that can be modeled by a parabola.
In the illustration, the unit of measurement for both axes is feet, and the vertex of the curve is point C Find a
quadratic function that models the hammock’s netting, and state the function’s domain.
The parent function is a quadratic. f ( x) = x2 . We need to find the parameters that change it into our model.
2
 1

We’ll use f ( x) = ±a ⋅  ± ( x − h)  + k . Because it is squared, the b value can be combined with a, and left out.
 b

So f ( x) = ±a ⋅ ( x − h ) + k . The vertex of the parent function is at (0, 0) so a shift horizontally h and vertically k
would put the vertex at (h, k). Our vertex is (3, 3) so we know the values of h and k. We can use these, as well
as one other coordinate for (x, y) and solve the basic equation for a.
2
4 = a ⋅ ( 8 − 3) + 3
2
4 = 25a + 3
1 = 25a
a=
1
25
1
2
( x − 3) + 3
25
Because this is a real-world situation, we need to restrict the domain. The endpoints of the hammock are at (-2,
4) and (8, 4). The domain is therefore restricted to {x | −2 ≤ x ≤ 8} or [-2,8]
Now we know that our hammock’s equation is f ( x) =
```
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