The quantities represented by a graph Graph the points (4, 9) and (7, 19) on a set of coordinate axes and sketch the straight line segment which runs from the first point to the second. Then sketch a straight vertical line segment from the first point down to the horizontal axis, and another from the second point down to the horizontal axis. Describe your sketch. Your sketch should consist of a point with horizontal coordinate 4 and vertical coordinate 9, and another with horizontal coordinate 7 and vertical coordinate 19. The line segment connecting these points slopes upward into the right. The first vertical line segment runs from the point (4, 9) to the point (4, 0) on the horizontal axis, and the second vertical segment runs from the point (7, 19) the point (7, 0) on the horizontal axis. Along with the segment which runs along the horizontal axis from (4, 0) to (7, 0), and these line segments form a trapezoid with vertical 'altitudes' of 9 and 21, and a 'width' of (7 - 4) = 3. (What we call the altitudes here are actually the bases of the trapezoid, and what we call the width is actually the altitude of the trapezoid, when using standard geometric terminology. However when referring to trapezoids of this nature on graphs, it makes more sense to referred to the vertical line segments as altitudes.) What is the rise between the points (4, 9) and (7, 21)? The second coordinate of an ordered pair is traditionally graphed on the ' vertical' axis, and the difference from the first vertical coordinate to the second is called the 'rise'. The 'rise' between these points is (21 - 9) = 12. What is the run between the points (4, 9) and (7, 21)? The first coordinate of an ordered pair is traditionally graphed on the 'horizontal' axis, and the difference from the first horizontal coordinate to the second is called the 'run'. The 'run' between these points is (7 - 4) = 3. What therefore is the slope between these points? As seen above the rise is 12 and the run is 3, so the slope is slope = rise / run = 12 / 3 = 3. What is the area of the trapezoid defined by these points? The area of a 'graph trapezoid' is the same as that of a rectangle having the same width, whose altitude is equal to the midpoint altitude of the trapezoid. The midpoint altitude of the trapezoid is simply the average of its two altitudes. For this reason we say that the midpoint altitude of the trapezoid is the average altitude of the trapezoid. The two altitudes of this trapezoid are 9 and 21, so the average altitude is their average (9 + 21) / 2 = 30 / 2 = 15. The width of the trapezoid can be measured from the point (4, 0) to the point (7, 0) and is 7 - 4 = 3 (you may also note that this is the same as the 'run' between the two points). The area of a rectangle is the product of its length and width, so the area of this trapezoid (being equal to the area of the rectangle) is therefore area = ave altitude * width = 15 * 3 = 45. What is the rise between the points (4 sec, 9 cm) and (7 sec, 21 cm)? The second coordinate of an ordered pair is traditionally graphed on the ' vertical' axis, and the difference from the first vertical coordinate to the second is called the 'rise'. The 'rise' between these points is (21 cm - 9 cm) = 12 cm. What is the run between the points (4 sec, 9 cm) and (7 sec, 21 cm)? The first coordinate of an ordered pair is traditionally graphed on the 'horizontal' axis, and the difference from the first horizontal coordinate to the second is called the 'run'. The 'run' between these points is (7 sec - 4 sec) = 3 sec. What therefore is the slope between these points? As seen above the rise is 12 cm and the run is 3 sec, so the slope is slope = rise / run = 12 cm / (3 sec) = 3 cm/sec. What is the area of the trapezoid defined by this graph? The area is equal to the average altitude of the trapezoid multiplied by its width. The altitudes of this trapezoid are 9 cm and 21 cm, so the average altitude is (9 cm + 21 cm) / 2 = 15 cm. The width is (7 sec - 3 sec) = 4 sec. So its area is 15 cm * 4 sec = 60 cm * s. Note that the unit cm * s is an essential aspect of the area and cannot be omitted. Note also that the unit cm * s has no general interpretation in terms of motion. It is not be confused with the unit cm / s, which is the unit of the slope (obtained by dividing a quantity with unit cm by a quantity with unit s). What is the rise between the points (4 sec, 9 cm / sec) and (7 sec, 21 cm / sec)? The second coordinate of an ordered pair is traditionally graphed on the ' vertical' axis, and the difference from the first vertical coordinate to the second is called the 'rise'. The 'rise' between these points is (21 cm / sec - 9 cm / sec) = 12 cm / sec. What is the run between the points (4 sec, 9 cm / sec) and (7 sec, 21 cm / sec)? The first coordinate of an ordered pair is traditionally graphed on the 'horizontal' axis, and the difference from the first horizontal coordinate to the second is called the 'run'. The 'run' between these points is (7 sec - 4 sec) = 3 sec. What therefore is the slope between these points? As seen above the rise is 12 cm / sec and the run is 3 sec, so the slope is slope = rise / run = (12 cm /s) / (3 sec) = 3 (cm/sec) / sec = 3 (cm/s) * (1/s) = 3 (cm * 1) / (s * s) = 3 cm / s^2. What is the area of the trapezoid defined by this graph? The area is equal to the average altitude of the trapezoid multiplied by its width. The altitudes of this trapezoid are 9 cm/s and 21 cm/s, so the average altitude is (9 cm/s + 21 cm/s) / 2 = 15 cm/s. The width is (7 sec - 3 sec) = 4 sec. So its area is 15 cm/s * 4 sec = 60 cm/s * s = 60 cm ( s / s) = 60 cm. Note that the product of 15 cm/s and 4 s can be interpreted as the product of an average velocity of 15 cm/s and the time interval of 4 s, so that with this interpretation the area represents the corresponding 60 cm displacement. In general, if our graph represents velocity versus clock time for an interval, the area under the graph represents the corresponding displacement. What is the rise between the points (4 ft, 9 lb) and (7 ft, 21 lb)? The second coordinate of an ordered pair is traditionally graphed on the ' vertical' axis, and the difference from the first vertical coordinate to the second is called the 'rise'. The 'rise' between these points is (21 lb - 9 lb) = 12 lb. What is the run between the points (4 ft, 9 lb) and (7 ft, 21 lb)? The first coordinate of an ordered pair is traditionally graphed on the 'horizontal' axis, and the difference from the first horizontal coordinate to the second is called the 'run'. The 'run' between these points is (7 ft - 4 ft) = 3 ft. What therefore is the slope between these points? As seen above the rise is 12 lb and the run is 3 ft, so the slope is slope = rise / run = 12 lb / (3 ft) = 3 lb/ft. What is the area of the trapezoid defined by this graph? The area is equal to the average altitude of the trapezoid multiplied by its width. The altitudes of this trapezoid are 9 lb and 21 lb, so the average altitude is (9 lb + 21 lb) / 2 = 15 lb. The width is (7 ft - 3 ft) = 4 ft. So its area is 15 lb * 4 ft = 60 lb * ft. Note that the product of 15 lb and 4 ft can be interpreted as the product of an average force of 15 lb and a displacement of 4 ft. As you will see later in the course this is a very important quantity, related to work and energy. You don't have to remember this interpretation, but you should understand that the units of the quantities represented on a graph determine the units of its slope and area, and that the meanings of the quantities represented on a graph determine the meanings of the slope and area. In some situations that can be difficult to understand meanings and interpretations, but the first step in the process is to include the units in every calculation and to do the algebra of the units. In general what does the rise between two points of a graph represent? The rise represents the change in the quantity represented by the vertical coordinate. In general what does the run between two points of a graph represent? The run represents the change in the quantity represented by the horizontal coordinate. In general what does the slope between two points of a graph represent? The slope represents the change in the quantity represented by the vertical coordinate, divided by the change in the quantity represented by the horizontal coordinate. If the graph represents quantity A vs. quantity B, then the slope between two points represents change in A / change in B, which is by definition the average rate at which quantity A changes with respect to quantity B. In general what does the average of the vertical quantities represented by two points represent? If the graph is a straight line, this average represents the average value of the quantity represented by the vertical coordinate. If the graph for the interval between the points is nearly a straight line, then this average is near the average value of the quantity, and hence represents the approximate average value of the quantity represented by the vertical coordinate. In general what does the width of the trapezoid defined by two points represent? The width is the same as the 'run' between the two points and represents the change in the quantity represented by the horizontal coordinate. In general what does the area of the trapezoid defined by two points represent? The area represents the product of the average 'graph altitude' and the width of the trapezoid. If it makes sense to multiply the average value of the 'vertical' quantity by the change in the 'horizontal' quantity, then the area of the trapezoid is probably an important quantity.