Linear Functions: Point-Slope Form
- Author:
- Whit Ford
Use this page to explore how the constants h, k, and m affect the appearance of a line whose equation is written in Point-Slope form:
Use each of the sliders to change the values of h, k, and m, and see how each parameter affects the appearance of the graph. Once you have a feel for the effects of each parameter, see if you can complete the challenges described below the graph.
Drag the points h, k and m along the green sliders at the bottom center of the graph and see what happens. Can you:
- Make the line go through the origin?
- Make the line go through the point (-3,-1)?
- Make the line move down as it goes to the right?
- Make the line horizontal (parallel to the x-axis)?
- Make the line vertical (parallel to the y-axis)?
Which of the above could NOT be achieved? Why not?
Looking at the equation
y = m(x - h) + k
Why does k have the effect it has? Compare how this function would look after distributing the m and collecting like terms with and without k.
If you set h to 0 (do this on the h slider above), what happens when you simplify the original equation?
y = m(x-0) + k
y = m(x) - m(0) + k
y = mx + k
Do you recognize the this form?
When h is 0, k becomes the y-intercept of the line. What happens to the line as k is changed?
If you set k to 0 (do this on the k slider above), what happens when you simplify the original equation?
y = m(x- h) + 0
y = m(x) - m(h)
y = mx + a constant
Gee... this looks like slope-intercept form again. So, if k is zero, what is the y-intercept? If both h and k are not zero, what will the y-intercept be?
What happens to the line as h is changed? Why does h have this effect?
Notice that you cannot tell, just by looking at two lines with the same slope, whether the first line was moved horizontally or vertically to produce the second line... so a given translation (shift) of a line can be produced by either a horizontal translation, or a vertical translation, or a little of both.
What happens to the line as m is changed? Why does m have this effect? Describe what is it about the way m is connected to x in the equation that causes it to have this effect.
Observe the distance between points Y and P as m is changed. Why does this distance stretch and shrink as it does?
If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/