Illustrates how Point-Slope and Slope-Intercept line formulas work.

In calculus, we often use the [i]tangent line[/i] to the function for analysis. As you'll see, usually we can find the slope from the function, and we know the one point where we want to place the tangent line. Thus, the point-slope method of finding the line equation will be used much more than the more-familiar slope-intercept form. This app illustrates how the point-slope formula works and how to change it to slope-intercept if need be.[br][br]We start by noting that the slope of a line passing through any two points [math]P_1=(x_1, y_1)[/math] and [math]P_2=(x_2, y_2)[/math] can be found as [math]m=\frac{y_2-y_1}{x_2-x_1}[/math]. If we multiply this equation on both sides by [math]x_2-x_1[/math], and switch the sides, we get [math]y_2-y_1=m(x_2-x_1)[/math]. But let's allow [math](x_2, y_2)[/math] to be any point [math](x,y)[/math]. This gives us the [b]point-slope formula[/b] [math]y-y_1=m(x-x_1)[/math]. We can then change this to the slope-intercept form by solving for [math]y[/math]: [math]y=mx+(y_1-mx_1)[/math]. Note that the quantity in the parentheses is "[math]b[/math]", the [math]y[/math]-intercept.[br][br]To begin, leave the three check boxes checked. One or more can be turned off to hide parts of the display for clarity.[br][br]The two purple dots are the points [math](x_1, y_1)[/math] and [math](x_2, y_2)[/math]. You can drag them anywhere on the graph. As you do, you can see how the slope is determined, by the difference in the points' [math]y[/math]-values divided by the difference in their [math]x[/math]-values ([math]m=\frac{\Delta y}{\Delta x}[/math]). This is illustrated by the dashed brown lines and the black numbers next to them. In purple, the value of the ratio is shown.[br][br]When we convert to the slope-intercept form, we saw above that [math]y=mx+(y_1-mx_1)[/math]. The part in parentheses gives us [math]b[/math], the [math]y[/math]-intercept value. It says that the [math]y[/math]-intercept will be at the [math]y[/math]-value of [math]P_1[/math], minus a distance given by the slope [math]m[/math] times the [math]x[/math]-coordinate of [math]P_1[/math]. But [math]x_1[/math] is just the "run" distance [math]\Delta x[/math] from the [math]y[/math]-axis to [math]P_1[/math]. Since [math]m=\frac{\Delta y}{\Delta x}[/math], we can solve for [math]\Delta y=m \Delta x[/math]. So we would expect the [math]y[/math]-intercept to be [math]m \Delta x[/math] up or down from [math]P_1[/math]'s [math]y[/math]-value: [math]b=y_1-mx_1[/math]. This is illustrated by the blue and red dotted arrows.