1.2 Slopes and Intercepts Summary:
This is basically the ratio between two points (X1, Y1) and (X2, Y2). Given these two points, you can subtract Y1 from Y2 [Y2-Y1]. The same thing is done for the X values, [X2-X1]. The variable used in formulas for slope is ‘m’, Such as in “y = mx + b”,‘m’ stands for slope.
Another way of calculating the slope is using the ratio “Rise/Run”. Given two points, you pick one point (generally the lower one). Then you count how many points you rise (move up) and divide that by how many points you have to go to the left or right.
If you have to move up and to the right to get to the other point, the slope, “m” is positive.
If you have to move up and to the left to get to the other point, the slope, “m” is negative. \
Example (Page 13):
Method #1: Formula: (1-4)/(3-0)= -3/3 = -1. The slope of this line is -1.
Method #2: Rise/Run: Rise up 3 [+3]. Run over to the left 3 [-3]. 3/-3= -1.
Either way, it is the same answer, m = -1.
The intercepts are when a line intersects an axis. There is an “X-intercept” and a “Y-intercept”. In the equation, y = mx + b, ‘b’ stands for the Y-intercept. Since the value of ‘x’ on the Y-axis is always zero, you could substitute ‘0’ for ‘X’ to find where it intersects the Y-axis.
When given just “b” and one other point, it is possible to find the equation of a line. Whenever ‘b’ is given, it means (0,__). The x value will always be zero, but the y-value will fluctuate based on where it actually intersects.
There are multiple ways to write an equation for a line. One of them is “Slope-Intercept Form”.
Y = mx + b.
‘m’ stands for the slope.
‘b’ stands for the y-intercept.
Examples: y = 4x + 6
y = -7x + 14
y = ½x+76
The higher the value of ‘m’, the steeper it is. The lower the value, the flatter it is.
It is possible to change the y-intercept (‘b’) but keep the slope (‘m’) the same. These are called parallel lines.
Apart from slope-intercept form, there is another way to write equations. This is known as “Standard form”.
Ax + By = C.
In this equation, A and B both can’t be zero.
Example (page 16)
To graph this type of equation, first substitute in 0 for ‘X’. You end up with y = -2. This is your y-intercept, (0,2). To find another point, find the x-intercept. Plug in 0 for ‘Y’. This results in x = 3. This is your second point, (3,0). Plot these two points and draw a line through them.
A horizontal line ( ) has a defined slope of zero.
A vertical line ( ) has an undefined slope.