Day: May 14, 2018

Pre-Calculus 11 Week 12 – Graphing Absolute Values as a Linear Function

alexr2015

This week was absolute values! An absolute value describes how far a number is from zero. When a question asks for an absolute value, it looks like this: |6|

Not that hard. How far is the number six from zero? Well, six. What about |-5|?

|-5| = 5 because -5 is five away from zero. That’s all there is to it.

Now, how do we graph an absolute value? A rule that should be remembered is that there will never be a part of your graph below the x-axis. If your graph is below the x-axis, that means your number is negative, which is wrong. Because absolute values are never negative.

Below is a table and a graph for the function y = x + 1 and y = |x + 1|.

See how the line practically bounces off of the x-axis? This is how an absolute value looks when it is graphed. The red line represents y = |x + 1| and the black line shows what the line would look like if we weren’t looking for the absolute value. The black line is y = x + 1.

Pre-Calculus 11 Week 13 – Graphing Reciprocal Functions

alexr2015

Aight, this is really crazy, you’re gonna wanna strap in.

Imagine this function. It’s an easy one. f(x) = x

Nothing you haven’t seen before. The y-intercept is even zero. Easy peasy. Now this graph  is probably rustling your jimmies, and that abomination is what we call a reciprocal function. In fact, this is the reciprocal of f(x) = x. That

one in particular looks like this: f(x) = \frac{1}{x}

Quick reminder that when we’re asked to reciprocate something, it means the numerator and denominator switch places. In every whole number, there’s always a number 1 as it’s denominator. It’s just invisible. So when we’re asked to reciprocate 3, it becomes 1/3. In the case when we’re asked to reciprocate a function, the whole thing becomes a denominator. So to reciprocate f(x) = x+3, it looks like f(x) = \frac{1}{x+3}

Also good to know is that those bendy lines that are defying the laws of everything you know are called hyperbolas.

So you’re probably wondering: how the hell am I supposed to graph that thing? Well, good question! First, we’re gonna pretend that it’s not a reciprocal function. Let’s attempt to graph f(x) = x Just a diagonal line, nothing special. Now, we’re gonna label some special points.

We need the asymptote, which is an invisible line/fence/great wall that the graph can never cross over. There’s gonna be at least one for x and one for y. There can be more than one on the x-axis, but that’s for another blog post. The asymptote is going to be wherever the line crosses the x-axis. In this case, the asymptote is going to be x = 0. For the y-axis, the asymptote is always going to be y = 0. Don’t fight it, it just is. It’ll change in grade 12, but don’t worry about it. The x-axis asymptote is the red vertical line.

Now the last thing we need to find are the other special points the invariant points. To find the invariant points, take y = 1 and y = -1 and find wherever it would be on your line. I’ve marked them in green on the graph. In the case of this line, our invariant points are (1,1) and (-1,-1).

Now from here, draw your hyperbola, make sure it intersects with your cool dude points, and WHATEVER YOU DO, DON’T LET THE CURVE CROSS THE ASYMPTOTE. Now, I love accuracy as much as the next pre-calc nerd, but as long as your teacher sees a curve, and that you haven’t crossed the asymptote, your graph is golden. Draw a t-chart if it makes you feel any better, but generally, this is what most reciprocal graphs are gonna look like. They’re translate to the left and to the right if their b value, or the y-intercept is higher or lower, and the hyperbolas can be farther or closer to each other depending on the slope.