Each line has a point where .
For , the corresponding y-coordinate would be
For , the corresponding y-coordinate would be
slope
Slope is a measurement of the tilt of a line.
If and are two points on a line, then the slope can be calculated as
is a common name for slope in linear equations.
Slope tells us how a line is tilted.
Suppose we have a line that contains the point and slope . Let represent any other point on the line. Then,
This can be rewritten as . We call this the point-slope form of a line. This can be
rewritten as , which can be viewed as a formula for a function:
Let be a linear function. Then, has a formula of the form . is called the rate-of-change of .
Suppose we have the Cartesian plane displaying the graph of with the vertical axis, , representing . Then we have , which gives us , which is a linear equation, which is the equation for the plotted curve - a line in this case.
This algebraic - geometric connection connects many algebraic characteristics with graphical features.
Parallel lines are lines with the same slope. They are graphs of linear functions with the same rate of change.
Let and .
Below are the graphs of and .
If two linear functions have the same rate of change, then they must differ by only a constant.
In the example above, or .
Graphically, this means that either line can be shifted vertically by units to land on the other.
Prependicular lines are lines that form a right angle.
Below are the graphs of two lines, and , whch intersect at the point .
The equation of line would be and would have the equation , where and are their respective slopes.
Each line has a point where .
For , the corresponding y-coordinate would be
For , the corresponding y-coordinate would be
Since we have a right angle, these points are corners of a right triangle. Therefore, the lengths have to satisfy the Pythagorean Theorem.
The Pythagorean Theorem gives us
The slopes are negative reciprocals of each other.
Almost every line intersects both the horizontal and vertical axes. These points are called the intercepts of the line.
If we think of the line as the graph of a linear function, , then the horizontal intercept looks like , which means and is the zero of . In this case, our formula looks like and is a factor of the formula.
The trio
all represent the same idea.
The vertical intercept looks like , where . This is not as valuable as the horizontal
intercept, which corresponds to a zero and a factor.
Horizontal lines have equations of the form , where the is the name of the vertical axis. You can view this equation as to see that every point on this line is of the form , making it horizontal.
The equation of this line is .
Every point on this line is of the form . It is the graph of the constant function .
Vertical lines have equations of the form , where the is the name of the horizontal axis. You can view this equation as to see that every point on this line is of the form , making it vertical.
The equation of this line is .
Every point on this line is of the form . This graph corresponds to no linear function, because the single domain number is paired with more than one codomain number.
Every line is the graph of a linear function, except vertical lines.
We have many forms for writing linear equations and formulas for linear functions.
Point-Slope Form
Given a point on a line, , and the slope, , we can write the formula as . This emphasizes the behavior around .
Slope-Intercept Form
If we choose our point to be the vertical intercept , then the slope-intercept form looks like or .
These are both sums.
However, our interest focuses on zeros of the function, which we would like to correspond to factors in our formulas.
As long as , we can take our sum and transform it into a product.
Since is linear, has one zero. It is . It is visually encoded as the horizontal intercept, .
This zero will always exist as long as .
Let .
is the only root of . We can rewrite this formula in factored form knowing that has to be a factor.
.
Since is the only root of , we know that is the only -intercept of the corresponding line. In either form, the coefficient of is . This coefficent measures the rate of change of and the slope of the line.
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more examples can be found by following this link
More Examples of Linear Functions