Use this page to explore how the constants [b]h[/b], [b]k[/b], and [b]m[/b] affect the appearance of a line whose equation is written in Point-Slope form:[br][br][math]y = m(x - h) + k[/math][br][br]Use each of the sliders to change the values of [b]h[/b], [b]k[/b], and [b]m[/b], 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:[br]- Make the line go through the origin?[br]- Make the line go through the point (-3,-1)?[br]- Make the line move down as it goes to the right?[br]- Make the line horizontal (parallel to the x-axis)?[br]- Make the line vertical (parallel to the y-axis)?[br] [br]Which of the above could NOT be achieved? Why not?[br] [br]Looking at the equation[br] y = m(x - h) + k[br]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.[br] [br]If you set h to 0 (do this on the h slider above), what happens when you simplify the original equation?[br] y = m(x-0) + k[br] y = m(x) - m(0) + k[br] y = mx + k[br]Do you recognize the this form? [br][br]When h is 0, k becomes the y-intercept of the line. What happens to the line as k is changed?[br][br]If you set k to 0 (do this on the k slider above), what happens when you simplify the original equation?[br] y = m(x- h) + 0[br] y = m(x) - m(h)[br] y = mx + a constant[br]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? [br] [br]What happens to the line as [b]h[/b] is changed? Why does [b]h[/b] have this effect?[br] [br]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.[br][br]What happens to the line as [b]m[/b] is changed? Why does [b]m[/b] have this effect? Describe what is it about the way [b]m[/b] is connected to [i]x[/i] in the equation that causes it to have this effect.[br][br]Observe the distance between points Y and P as [b]m[/b] is changed. Why does this distance stretch and shrink as it does?[br][br]If you wish to use other applets similar to this, you may find an index of all my applets here: [url=https://mathmaine.wordpress.com/2010/04/27/geogebra/]https://mathmaine.com/2010/04/27/geogebra/[/url]