Understanding how changes in the equation of a function result in **stretching** and/or **reflecting** the graph of the function is a great way to take some of the mystery out of graphing more complicated equations. By recognizing the family to which a more complex equation belongs, and then identifying what changes have been made to the parent of that family, the graph of even quite detailed functions can be made much more understandable.

See if you can identify what parts of the equation: \begin{align*}y = -\frac{1}{5} x^{2}\end{align*}

### Stretching and Reflecting Transformations

#### Stretching and Compressing Graphs

If we multiply a function by a coefficient, the graph of the function will be stretched or compressed.

Given a function f(x), we can formalize compressing and stretching the graph of f(x) as follows:

- A function
*g*(*x*) represents a vertical stretch of*f*(*x*) if*g*(*x*) =*cf*(*x*) and*c*> 1.

- A function
*g*(*x*) represents a vertical compression of*f*(*x*) if*g*(*x*) =*cf*(*x*) and 0 <*c*< 1.

- A function h(x) represents a horizontal compression of
*f*(*x*) if*h*(*x*) =*f*(*cx*) and*c*> 1.

- A function
*h*(*x*) represents a horizontal stretch of*f*(*x*) if*h*(*x*) =*f*(*cx*) and 0 <*c*< 1.

Notice that a vertical compression or a horizontal stretch occurs when the coefficient is a number between 0 and 1.

#### Reflecting Graphs Over the y-axis and x-axis

Consider the graphs of the functions *y* = *x*^{2} and *y* = -*x*^{2}, shown below.

The graph of *y* = -*x*^{2} represents a **reflection** of *y* = *x*^{2}, over the *x*-axis. That is, every function value of *y* = -*x*^{2} is the negative of a function value of *y* = *x*^{2}. In general, *g*(*x*) = -*f*(*x*) has a graph that is the graph of *f*(*x*), reflected over the *x*-axis.

### Examples

#### Example 1

Earlier, you were asked a question about identifying transformations.

The function \begin{align*}y = -\frac{1}{5}x^{2}\end{align*}*x* axis, because of the negative co-efficient on the *x*, and vertically compressing it (making it *wider*), because the co-efficient on the *x* is a fraction between 0 and 1.

#### Example 2

Identify the graph of the function *y* = (3*x*)^{2}.

We have multiplied *x* by 3. This should affect the graph horizontally. However, if we simplify the equation, we get *y* = 9*x*^{2}. Therefore the graph if this parabola will be taller/thinner than *y* = *x*^{2}. Multiplying *x* by a number greater than 1 creates a horizontal compression, which looks like a vertical stretch.

#### Example 3

Identify the transformation described by *y* = ((1/2)*x*)^{2}.

If we simplify this equation, we get *y* = (1/4) *x*^{2}. Therefore multiplying *x* by a number between 0 and 1 creates a horizontal stretch, which looks like a vertical compression. That is, the parabola will be shorter/wider.

#### Example 4

Sketch a graph of *y* = *x*^{3} and y = -*x*^{3} on the same axes.

At first the two functions might look like two parabolas. If you graph by hand, or if you set your calculator to sequential mode (and not simultaneous), you can see that the graph of *y* = -*x*^{3} is in fact a reflection of *y* = *x*^{3} over the x-axis.

However, if you look at the graph, you can see that it is a reflection over the *y*-axis as well. This is the case because in order to obtain a reflection over the *y*-axis, we negate *x*. In other words, *h*(*x*) = *f*(-*x*) is a reflection of *f*(*x*) over the *y*-axis. For the function *y* = *x*^{3}, *h*(*x*) = (-*x*)^{3} = (-*x*) (-*x*) (-*x*) = -*x*^{3}. This is the same function as the one we have already graphed.

It is important to note that this is a special case. The graph of *y* = *x*^{2} is also a special case. If we want to reflect *y* = *x*^{2} over the *y*-axis, we will just get the same graph! This can be explained algebraically: *y* = (-*x*)^{2} = (-*x*) (-*x*) = *x*^{2}.

#### Example 5

Graph the functions \begin{align*}y = \sqrt{x}\end{align*}

The equation \begin{align*}y = \sqrt{-x}\end{align*}*x* under the square root. It is important to keep in mind that - *x* means the *opposite* of *x*. Therefore the domain of this function is restricted to values ≤ 0. For example, if *x* = - 4, \begin{align*}y = \sqrt{-(-4)}= \sqrt{4} = 2\end{align*}. It is this domain, which includes all real numbers not in the domain of \begin{align*}y = \sqrt{x}\end{align*} plus zero, that gives us a graph that is a reflection over the *y*-axis.

In sum, a graph represents a reflection over the *x*-axis if the function has been negated (i.e. the *y* has been negated if we think of *y* = *f*(*x*)). The graph represents a reflection over the *y*-axis if the variable *x* has been negated.

#### Example 6

Identify the function and sketch the graph of \begin{align*}y = \sqrt{x}\end{align*} reflected over both axes.

To reflect the graph of \begin{align*}y = \sqrt{x}\end{align*} over both axes, the function must be *negated* both outside and inside the root: \begin{align*}y = -\sqrt{-x}\end{align*}. The negation (negative) *outside* of the root has the effect of reflecting the graph vertically, and the negation *inside *of the root reflects the graph horizontally. The image below shows three versions:

BLUE: \begin{align*}y = \sqrt{x}\end{align*}

GREEN: \begin{align*}y = -\sqrt{x}\end{align*}

RED: \begin{align*}y = \sqrt{-x}\end{align*}

### Review

- If a function is multiplied by a coefficient, what will happen to the graph of the function?
- What does multiplying x by a number greater than one create?
- What happens when we multiply x by a number between 0 and 1
- In order to obtain a reflection over the y axis what do we have to do to x?
- How do we obtain a reflection over the x-axis?
- Write a function that will create a horizontal compression of the following:\begin{align*}f(x) = x^2 + 3\end{align*}
- Write a function that will horizontally stretch the following: \begin{align*}f(x) = x^2 - 6\end{align*}
- Rewrite this function\begin{align*}f(x) = -\sqrt{x}\end{align*}to get a reflection over the x-axis.
- Rewrite this function\begin{align*}f(x) = \sqrt{x}\end{align*}to get a reflection over the y-axis.

Graph each of the following using transformations. Identify the translations and reflections.

- \begin{align*}f(x) = |x| -2\end{align*}
- \begin{align*}h(x) = \sqrt{x + 3}\end{align*}
- \begin{align*}g(x) = \frac{1}{x + 1}\end{align*}
- \begin{align*}f(x) = -4x^3\end{align*}
- \begin{align*}h(x) = (x + 3)^3 + 1\end{align*}
- \begin{align*}f(x) = \frac{1}{3}(x - 3)^2 + 1\end{align*}
- \begin{align*}f(x) = -4\sqrt{x + 1} - 2\end{align*}
- \begin{align*}f(x) = \frac{2}{3(x - 2)} + \frac{1}{4}\end{align*}

Let \begin{align*}y = f(x)\end{align*} be the function defined by the line segment connecting the points (-1, 4) and (2, 5). Graph each of the following transformations of \begin{align*} y = f(x)\end{align*}.

- \begin{align*}y = f(x) + 1\end{align*}
- \begin{align*}y = f(x+ 2)\end{align*}
- \begin{align*}y = f(-x)\end{align*}
- \begin{align*}y = f(x + 3) - 2\end{align*}

The graph of y = x is shown below. Sketch the graph of each of the following transformations of y = x

- \begin{align*}y = x + 3\end{align*}
- \begin{align*}y = x - 2\end{align*}
- \begin{align*}y = - x\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.13.