We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

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Transformations of Trigonometric Functions

The transformation of functions includes the shifting, stretching, and reflecting of their graph. The same rules apply when transforming trigonometric functions.

Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph of

$y = f(x) + c$: shift the graph of $y = f(x)$ up by $c$ units

$y = f(x) - c$: shift the graph of $y = f(x)$ down by $c$ units

$y = f(x - c)$: shift the graph of $y = f(x)$ to the right by $c$ units

$y = f(x + c)$: shift the graph of $y = f(x)$ to the left by $c$ units

Example: Sketch the function $y=\sin(x), y=\sin(x)+4,$ and $y=\sin(x+\frac{\pi}{2})$.

Solution:

$graphs of sin(x) with vertical and horizontal shifts$

Here, $y=\sin(x)$ is shown in black. The curve $y=\sin(x)+4$ is shifted 4 units up, shown in green. The curve $y=\sin(x+\frac{\pi}{2})$ is shifted $\frac{\pi}{2}$ units to the left, and is shown in blue. Notice that this is the same thing as the curve $y=\cos(x)$.

Vertical and Horizontal Stretches/Compressions

Suppose c > 1. To obtain the graph of

$y = cf(x)$: stretch the graph of $y = f(x)$ vertically by a factor of $c$

$y = \frac{1}{c} f(x)$: compress the graph of $y = f(x)$ vertically by a factor of $c$

$y = f(cx)$: compress the graph of $y = f(x)$ horizontally by a factor of $c$

$y = f(\frac{x}{c})$: stretch the graph of $y = f(x)$ horizontally by a factor of $c$

Example: Sketch the functions $y=\cos(x), y=\cos(2x),$ and $y=3\cos(2x)$.

Solution:

$cosine graph with horizontal compression and vertical stretch$

The function$y=\cos(x)$ is shown in black. The blue curve represents the function $y=\cos(2x)$ and is compressed horizontally by a factor of 2. The green curve represents the function $y=3\cos(2x)$ and is the blue curve stretched vertically by a factor of 3.

Reflections

To obtain the graph of

$y = -f(x)$: reflect the graph of $y = f(x)$ about the x-axis; and

$y = f(-x)$: reflect the graph of $y = f(x)$ about the y-axis

Example: Given that the blue curve represents the function $y=\sin(x)$, determine whether the following statement is TRUE or FALSE.

The green curve on the following graph represents both $y=-\sin(x)$ and $y=\sin(-x)$.

$sine graph reflected in x-axis and y-axis$

Solution: TRUE - The green curve does represent $y=-\sin(x)$ and $y=\sin(-x)$. In this particular situation the reflection about the x-axis is the same as the reflection about the y-axis.

Example 1:

Example 2: