Functions and transformations

Quadratic equations Functions and transformations

Functions can be used to describe transformations of the graph $y = f(x)$ .

$f(x) + a$ is the same as a translation of $a$ up and can be described by the vector $begin{pmatrix} 0 \ a end{pmatrix}$

Let’s use $f(x) = x^2 + 2x - 3$ as an example. Here is a graph of $y = f(x)$

Now we will graph $y = f(x) + 3$ which is the same as a translation of $begin{pmatrix} 0 \ 3 end{pmatrix}$

$f(x) - a$ is the same as $f(x) + (-a)$ and so is a move of $-a$ upwards, or in other words, a move of $a$ downwards. The translation can be described by the vector $begin{pmatrix} 0 \ -a end{pmatrix}$

$f(x + a)$ is a move left by $a$ . The translation can be described by the vector $begin{pmatrix} -a \ 0 end{pmatrix}$

Let’s use $f(x) = x^2 + 2x - 3$ as an example again. Then $f(x + 3) = (x+3)^2 + 2(x + 3) - 3$

$=(x+3)(x+3) + 2x + 6 - 3$

$=x^2 + 6x + 9 + 2x + 3$

$=x^2 + 8x + 12$

Below is a plot of $f(x)$ and $f(x+3)$

$f(x - a)$ is the same as $f(x + (-a))$ and so is a move of $-a$ left, or in other words, a move of $a$ right. The translation can be described by the vector $begin{pmatrix} a \ 0 end{pmatrix}$

$-f(x)$ is the same as a reflection in the x-axis.

Let’s use $f(x) = x^2 + 2x - 3$ as an example again. Then $-f(x) = -(x^2 + 2x - 3) = -x^2 - 2x + 3$

$f(-x)$ is the same as a reflection in the y-axis.

Let’s use $f(x) = x^2 + 2x - 3$ as an example again. Then $f(-x) = (-x)^2 + 2(-x) - 3 = x^2 - 2x - 3$

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