Further curve sketching (1 Viewer)

[Michael7]

Hi, could anyone please explain the reasoning to these sketches. I don't feel satisfied just memorising them, without knowing how they really work

y = -f(x) --> just change the y-coordinate of every point on the function y = f(x), i.e flip f(x) about the x-axis
y = f(-x) --> change the x-coordinate of every point on the original function
y = |f(x)| --> ?
y = f|(x)| ---> same y-value for the corresponding x-value. x>0 = f(x), x<0 = f(x)
|y| = f(x) --> this confuses me a lot

Thanks in advance!

 

[fan96]

Imagine the graph of any function on the Cartesian plane, for example .

The graph of a function is a visualisation of the relationship between an input and an output .

1)

This can be rewritten as . You're replacing with , swapping the sign of each output. On a graph, this means that becomes , etc. If every value on the graph has its sign reversed, then you've basically flipped it across the axis.

2)

Similar to 1), you are replacing with . If you have two points and on the graph, they will become and after the swap. This flips the graph across the axis.

3)
Think of the graph here. Because of the absolute value function, any negative values of will now become positive. Basically, you're flipping the graph (from down to up) across the axis ONLY where the value was negative before.

4)
Again, because of the absolute value function, any negative values of become positive before going into the function. So , , etc. You flip the graph from right to left across the axis.

5)
This is really tricky. It's really helpful to visualise this using an example. Look at the graphs of and .

Firstly, is only defined for if would have produced a positive number. doesn't exist if , for example. So any values where are removed.

So what happens when is positive? Well if , that means . So for an value that exists on the graph, the value could be either OR (incidentally, this means that is not a function). This is the same as mirroring the remaining graph across the x-axis.

To summarise: start with , exclude all values of for which , and mirror what's left across the x-axis.

 

[Michael7]

Thank you so much, i really appreciate it. Understood everything except for the last one. I'm a little confused

 

[fan96]

Okay, let me elaborate then. We can rewrite the relation to make it a bit less confusing.



Using one of the definitions of the absolute value function ,





(when we square both sides, we unintentionally define the relation for negative values of - to fix that, we can restrict the domain of the resulting relation)

We end up with



Which is equivalent to combining these two functions:




______________________________________________________________________

Basically, the takeaway is that

is the same thing as
______________________________________________________________________

That means for every (valid) value you input, the output value is both AND

Here's an example with