Graphs, graphs, graphs...

I was going to do a play by play of the chapters I've already done, but I'm afraid of forgetting my current epiphanies. I'll just back date those entries.


Today I covered chapters 5 and 6. Chapter 5 is the basics of graphing and graphing equalities. In a nut shell I learned graphs represent all possible solutions to an equation. I'm sure at one point I knew that, but today it was an "a-HA! My math teachers had a reason for this other than torturing me!" Basically instead of randomly choosing an x or y and plugging them in an equation -- my time honored method -- you can graph the equation and see which  x and y work together. I only see intercepts easily. Maybe that's all I really need to see well. Absolute value was a looooooooooong forgotten concept. Use it to find the vertex of a V-shaped graph. Got it. Only one of the example problems made me head tilt.

What coordinate pair represents the vertex of the graph of the equation y = -|x - 4| - 5 ?

First I solve the equation inside the absolute value for x. (I circle the original operation in my work so I don't confuse which one goes on both sides. Here it's highlighted in red.)

x - 4 = 0
x - 4 + 4 = 0 + 4
x = 4

So now

y = -|4 - 4| - 5
y = -|0| - 5

...what the !@#$ do I do with this negative sign outside the absolute value bars when my answer is 0? There's no negative zero like there's no crying in baseball! My pea brain decided since the 0 disappears I should be left with

y = - - 5
y = 5

The actual answer was y = -5. So the negative sign should've disappeared with the 0. Okay.

But why was that stupid negative sign there in the first place?


Chapter 6 was formula overload. I actually wrote them all in the first page of my fresh new notebook so I could keep track.

Formulas and concepts (re)learned

Slope formula

• (d-b) / (c-a) = m

• m = slope of line

• Positive slopes rise right to left.
• Negative slopes fall left to right.
• Horizontal slope is 0.
• Vertical slope is undefined because you can't divide by 0. Unless you get into that imaginary number stuff and I'm nowhere near the i chapter yet. That might even be a whole other book.

Point-slope formula

• y - y₁ = m(x - x₁)

• Good: Only one set of coordinates needed
• Bad: More opportunity for me to screw up multiplying all the way across the parentheses.

Slope-intercept formula

• y = mx + b

• b = y-intercept

• Good: x will always be 0!
• Bad: If there's a coefficient with y, there will be division. Must remember to divide on both sides of equation

I did all the practice problems. Most of my mistakes were forgetting to carry negative signs from one step to another, deciding 6/3 = 3, or just plain copying the equation wrong. Concepts solid, second grade math and reading comprehension sloppy. That's what my scientific calculator app is for.

Standard form of a line

This is the part where they tell me slope-intercept form is bad mathematical grammar. Correcting other's grammar in literary form with red ink is way more fun. *sigh* If the slope is a whole number it's golden. If the slope is a fraction I have my calculator ready. I solved this problem all by myself...until the last few steps.

Write the equation of the line that passes through the points (-3,7) and (4,1) in standard form.

This one uses all the goodies in the bag. I really do have to do every single step in the equation or I'll screw up on the shortcut. First calculate m.

m = (1 - 7) / [4 - (-3)]

m = (1 - 7) / (4 + 3)

m = -(6/7)

Now point-intercept. I chose (4,1) because they're both positive.

y - 1 = -(6/7)(x - 4)

y - 1 = -(6/7)x + [-(6/7) * (-4/1)]

y - 1 = -(6/7)x + (24/7)

(7)y -1(7) = [(7/1) * -(6/7)] + [(7/1) * (24/7)]

I am still adept enough with fractions to know when things cancel out.

7y - 7 = -6x + 24

Remember above when I said I circled the original operation in my work? This is the problem that made me start doing it. My original solution was

7y - 7 + 7 = -6x + 24 - 7 etc. y = 17

Flip to the back of the book. Not 17? Wait, 24 + 7 is...argh.

7y - 7 + 7 = -6x + 24 + 7

7y = -6x + 31

6x + 7y = 31

I got there. Wasn't pretty, but I got there.

My brain is a little fried from the endless formatting. My struggle with gleaning equations from lines perpendicular and parallel to the line you're looking for will have to wait until Thursday. My kid turns x - 5 = 0 tomorrow!

Did I just do that? I must be making progress!