Today's soundtrack is Christmas With the Mills Family, a Spotify playlist that includes my favourite Christmas songs.
This Christmas Eve morning, I'm starting the next chapter of this unit, "Graphing Linear Inequalities in Two Variables."
Right off the bat, I realized that I would need a refresher on linear graphs before tackling this unit, as I was given the equation "3x + 2y + 9 = 0" and told to "[i]dentify the slope, y-intercept, and x-intercept" (Pearson's Pre-Calculus 11, p. 354). I know that I learned this in last year's course, but I don't remember how to do it now.
So we head off... to the internet!
. . .
Okay, so using the slope-intercept form (y = mx + b), we can easily graph a linear equation. Here are the steps:
Ensure equation is in slope-intercept form (y = mx + b) by isolating y
In my case, I had to convert 3x + 2y + 9 = 0 to slope-intercept form. My first step was to move x and c to the right side of the equation, giving me 2y = -3x - 9. My next step was to isolate y by dividing the entire equation by 2, giving me y = -3/2x - 4.5
Identify the characteristics of the equation, then draw them on a graph
Our slope is m, and our y-intercept is b. So in my equation, y = -3/2x - 4.5, I know that my y-intercept is -4.5, so I'd draw that dot on my graph. My slope is -3/2; since my slope is negative, the line (dotted if < or >, solid if ≥ or ≤) will go down three and right two (or up three and left two). If the slope was positive, it would go up the value of the numerator and right the value of the denominator of m. If m was a whole number, we would read the implied denominator of 1.
One thing to note: this method doesn't make the x-intercept incredibly clear. To get the exact value of the x-intercept, substitute 0 for y, then solve for x. In my example, I started with 2y = -3x - 9. Substituting 0 for y gave me 2(0) = -3x - 9, which came to 0 = -3x - 9. I moved -3x to the left side of the equation, resulting in 3x = -9. To isolate x, I divided both sides of the equation by 3, giving me the answer: x = -3. So my x-intercept is -3.
Shade the correct region of the graph
If y > the equation, shade the area above the dotted line
if y < the equation, shade the area below the dotted line
If y ≥ the equation, shade the area above the solid line
if y ≤ the equation, shade the area below the solid line
Test at the origin to determine whether it is part of the solution
A graph's origin is (0, 0). Substitute 0 for both x and y, then determine whether the statement is true or not. If the statement (e.g. 0 < 10) is true, then the region including the origin should be shaded. If the statement is false, then the region including the origin should not be shaded.
That's enough math for today! Time to finish wrapping Christmas presents!