To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with.
Y and x stand for the coordinates of any points on the line. Remember that slope is the change in y or rise over the change in x or run.
Now, b gives us the value of y where x is zero, this is called the y-intercept or where the line will cross the y axis. Locate this point on the y axis. Your slope is the coefficient of your x term.
The numerator tells us that the y value for the next coordinate increase by 5, the denominator tells us that the x value for the next coordinate changes by 1, so we can add this values to our starting coordinate of 0, Now do the same process with the new point of 1, The next point is 2, 4.
Connect these three points and label to graph it correctly. Alternately, you could create a table of values, choosing values for x like -1, 0 and 1 and plugging them into the equation one at a time to solve for the corresponding x coordinate.
So your points would be Therefore, you need only two points.
Plug in 0 and 1 for x: There is your plot. Now go over to the right 6 now you have the point 6, See how the first example all positive goes up, left to the right and the second goes down, left to the right. Something to keep in mind while drawing your graph is that the larger the bottom, or run, is in relationship to the rise the closer the slope will be to the x-axis.
Next we are going to work with b. We know three points along this line -2,3-4,6and -6,9. Add 2 to each y making them -2,5-4,8and -6,To establish a rule for the equation of a straight line, consider the previous example. An increase in distance by 1 km results in an increase in cost of $3.
Practice finding the equation of a line passing through two points. Practice finding the slope-intercept equation of a line from its graph.
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YOUR TURN: Find the equation of the line passing through the points (-4, 5) and (2, -3). Method 1: Using Slope Intercept Form. What is the equation of line parallel to y = 3x + 5 and through the point (1, 7)?. Many students are more comfortable using slope intercept form but this kind of problem is actually much easier, using point slope form (which is right below this work).
Substitute the slope from original line (3 in this case) into the equation of the line y = 3x + b. The graph of f is shown below. Notes that 1) As x approaches 3 from the left or by values smaller than 3, f (x) decreases without bound. 2) As x approaches 3 from the right or by values larger than 3, f (x) increases without bound.