randRange(-9, 9) randRange(-9, 9) randRange(-9, 9) randRange(-9, 9) (Y1 - Y2) / (X1 - X2) randRange( 0, 1 )

What is the slope of the line through the points (X1, Y1) and (X2, Y2)?

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); line( [X1 - 19, Y1 - 19 * M], [X2 + 19, Y2 + 19 * M], { stroke: "#888" } ); style({ fill: ORANGE, stroke: ORANGE }); circle( [X1, Y1], 3/20 ); style({ fill: BLUE, stroke: BLUE }); circle( [X2, Y2], 3/20 );
(Y1 - Y2) / (X1 - X2)

Picture an airplane moving from left to right. If the the airplane is taking off \color{BLUE}{/}, the slope is positive. If it's landing \color{GREEN}{\backslash}, the slope is negative. When the plane has reached cruising altitude \color{ORANGE}{-}, the slope is 0.

graphInit({ range: 4, scale: 20, tickStep: 1, labelStep: false }); style({ fill: BLUE, stroke: BLUE }); line( [ -1,-1 ], [ 1,4 ] ); label([0,-4], "\\color{BLUE}{\\text{Airplane Taking Off}}", "below");
graphInit({ range: 4, scale: 20, tickStep: 1, labelStep: false }); style({ fill: GREEN, stroke: GREEN }); line( [ 0, 2 ], [ 2, -1 ] ); label([0,-4], "\\color{GREEN}{\\text{Airplane Landing}}", "below");

The quicker the airplane takes off, the steeper the positive slope, which means the answer will be a higher number than when the airplane takes off slowly. The quicker the airplane lands, the steeper the negative slope, which means the answer will be much more negative than when it lands slowly.

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); line( [X1 - 19, Y1 - 19 * M], [X2 + 19, Y2 + 19 * M], { stroke: "#888" } ); style({ fill: ORANGE, stroke: ORANGE }); circle( [X1, Y1], 3/20 ); style({ fill: BLUE, stroke: BLUE }); circle( [X2, Y2], 3/20 );

The equation for the slope is m = \dfrac{\color{BLUE}{y_2} - \color{ORANGE}{y_1}}{\color{BLUE}{x_2} - \color{ORANGE}{x_1}} for points (\color{ORANGE}{X1}, \color{ORANGE}{Y1}) and (\color{BLUE}{X2}, \color{BLUE}{Y2}).

style({ fill: "", stroke: PINK }); line( [ X1, Y2 ], [ X2, Y2 ] ); style({ stroke: GREEN }); line( [ X1, Y1 ], [ X1, Y2 ] );

Substituting in, we get m = \dfrac{\color{BLUE}{Y2} - \color{ORANGE}{negParens(Y1)}}{\color{BLUE}{X2} - \color{ORANGE}{negParens(X1)}} = \dfrac{\color{GREEN}{Y2 - Y1}}{\color{PINK}{X2 - X1}}

So, the slope m is fractionReduce( Y2 - Y1, X2 - X1 ).

[ { name: "Blue", hex: KhanUtil.BLUE }, { name: "Orange", hex: KhanUtil.ORANGE }, { name: "Green", hex: KhanUtil.GREEN }, { name: "Pink", hex: KhanUtil.PINK } ] randRange( 2, 5 ) [ { value: M_INIT, display: M_INIT }, { value: -1 * M_INIT, display: "-" + M_INIT }, { value: 1 / M_INIT, display: "\\dfrac{1}{" + M_INIT + "}" }, { value: -1 / M_INIT, display: "-\\dfrac{1}{" + M_INIT + "}" } ] randRange( -3, 3 ) randRange( 0, 3 ) [ 0, 1, 2, 3 ] SLOPES[WHICH]

Which graph best depicts a slope of M.display?

graphInit({ range: 6, scale: 20, tickStep: 1, labelStep: 1, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); style({ stroke: COLORS[index].hex }); label([0,-6], "\\color{" + COLORS[index].hex + "}" + "{\\text{" + COLORS[index].name + "}}", "below"); plot(function( x ) { return ( x - 1 ) * SLOPES[index].value + B; }, [ -11, 11 ]);
\quad \color{COLORS[WHICH].hex}{\text{COLORS[WHICH].name}}
1. \quad \color{COLORS[index].hex}{\text{COLORS[index].name}}

Remember that the slope corresponds to which direction the line slants, and how much it slants.

Because M.display is negative, the line should slant downwards as we follow it to the right.

The answer is either the \text{orange} or \text{pink} graph.

Because M.display is positive, the line should slant upwards as we follow it to the right.

The answer is either the \text{blue} or \text{green} graph.

In which graph does the y value change by M.display if the x value changes by 1?

The \color{COLORS[WHICH].hex}{\text{COLORS[WHICH].name.toLowerCase()}} graph best depicts a slope of M.display.

[ { value: M_INIT, display: M_INIT }, { value: 0, display: 0 }, { value: 99999, display: "undefined" }, { value: 1 / M_INIT, display: "\\dfrac{1}{" + M_INIT + "}" } ] randRange( 1, 2 ) SLOPES[WHICH]

Which graph best depicts a slope of M.display?Which graph best depicts an undefined slope?

Remember that the slope corresponds to how steep the line is.

Imagine walking up a hill represented by a line. A larger slope means a steeper hill.

A slope of M.display means there is no hill at all, and the graph should be flat.

An M.display slope corresponds to a vertical line, which is like an impossibly steep hill.

The \color{COLORS[WHICH].hex}{\text{COLORS[WHICH].name.toLowerCase()}} graph best depicts an undefined slope of M.display.