Transformations of Graphs - how to avoid the common mistakes

Concept: Transforming from graph of $y = f(x)$ to graph of $y = f(x - 1)$. Common mistake(s): Translating in the negative x direction instead of positive x direction. How to avoid the common mistake(s): Understand that when x is replaced with (x − 1), the value of x has to be increased by 1 to obtain the same value for y. For example, if:
y = f(x) = 5x x = 2 gives y = 10 x = −3 gives y = −15	y = f(x − 1) = 5(x − 1) x = 3 gives y = 10 x = −2 gives y = −15

Concept: Transforming from graph of $y = f(x)$ to graph of $y = f(2x)$. Common mistake(s): Scaling by factor of 2 instead of $\frac{1}{2}$. How to avoid the common mistake(s): Understand that when x is replaced with 2x, the value of x has to be halved to obtain the same value for y. For example, if:
y = f(x) = 5x x = 2 gives y = 10 x = −3 gives y = −15	y = f(2x) = 5(2x) x = 1 gives y = 10 x = $- \frac{3}{2}$ gives y = −15

Concept: Transforming from graph of $y = f(x)$ to graph of $y = f( - x)$. Common mistake(s): Reflecting about x-axis instead of y-axis. How to avoid the common mistake(s): Understand that when x is replaced with −x, the value of x has to be negated to obtain the same value for y. For example, if:
y = f(x) = 5x + 1 x = 2 gives y = 11 x = −3 gives y = −14	y = f(−x) = 5(−x) + 1 x = −2 gives y = 11 x = 3 gives y = −14

Concept: Getting from graph of $y = f(x)$ to graph of $y = f\left( |x| \right)$. Common mistake(s): Giving the graph of $y = f\left( |x| \right)$ instead; and Reflecting all parts of the graph where x < 0 about y-axis and retaining all parts of the graph where x > 0. How to avoid the common mistake(s): Understand that when x is replaced with |x|, x = −2, for example, gives the same value for y as x = 2. For example, if:

y = f(x) = 5x x = 2 gives y = 10 x = −3 gives y = −15

y = f(|x|) = 5|x| x = 2 gives y = 10 x = −2 gives y = 10 No value of x gives y = −15

Concept: Transforming from graph of $y = f(x)$ to graph of $y = f(2x - 1)$. Common mistake(s): Replacing x with 2x and then replacing x with (x – 1) instead of replacing x with (x – 1) and then replacing x with 2x. How to avoid the common mistake(s): Write out the replacement, i.e.: write out $y = f(2x)$ when replacing x with 2x; and then write out $y = f\left( 2(x - 1) \right) = f(2x - 2)$ when replacing x with (x – 1).

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