Slope as a Rate of Change

Transcription

1 Find the slope of a line using two of its points. Interpret slope as a rate of change in real-life situations. FINDING THE SLOPE OF A LINE The slope m of the nonvertical line passing through the points (, ) and (, ) is (, ) m = r ise = c hange run chang in e in = º. º (, ) Read as sub one. Think -coordinate of the first point. Read as sub two. Think -coordinate of the second point. CONCEPT SUMMARY CLASSIFICATION OF LINES BY SLOPE A line with positive A line with negative A line with zero A line with slope rises from left slope falls from left slope is horizontal. undefined slope to right. to right. is vertical. GOAL INTERPRETING SLOPE AS A RATE OF CHANGE m > 0 m < 0 m = 0 m is undefined. A rate of change compares two different quantities that are changing. For eample, the rate at which distance is changing with time is called velocit. Slope provides an important wa of visualizing a rate of change. EXAMPLE 6 Slope as a Rate of Change You are parachuting. At h

2 FINDING THE SLOPE OF A LINE Algebra! Name! 4.4 Notes The Finding slope mthe of Slope the nonvertical of a line line (pp 6-9) The passing slope through m of a nonvertical the points line (is, ) (, 7) 7 the and number (, of ) units is the line rises or (, ) falls for each unit of horizontal rise = 7 º 4 = units change m from = r ise left = to c hang right. (, 4) run chang e in e in = º. º (, ) The slanted line at the right rises units for each units of horizontal run = º = units change Read from as left to sub right. one. So, the Think -coordinate of the slope m of the line is. first point. Read as sub rise slope = r u n = 7 two. Think º 4 -coordinate of the = º second point. Two points on a line are all that is needed to find its slope. When ou use the formula for slope, the order of the subtraction is important. Given two points on a line, ou can label either point as (, ) and the other point as (, ). After doing this, however, ou must form the numerator and the denominator using the same order of subtraction. Correct: m = º º Incorrect: m = º º Subtraction order is the same. Subtraction order is different.

3 EXAMPLE A Line with a Positive Slope Rises EXAMPLE STUDENT HELP Look Back For help with evaluating epressions, see p. 79. Find the slope of the line passing through (º, ) and (, 4). SOLUTION Let (, ) = (º, ) and (, ) = (, 4). º m = º 4 º = º (º ) = + Rise: Difference of -values Run: Difference of -values Substitute values. Simplif. = Slope is positive. (, ) (, 4) (, ) (, 4) Positive slope: line rises from left to right. 7. Find the slope of a line passing through (-, 0) & (-, 6). Positive slope: line rises from left to right. EXAMPLE A Line with a Zero Slope is Horizontal Find the slope of the line passing through (º, ) and (, ). SOLUTION Let (, ) = (º, ) and (, ) = (, ). º m = º º = º (º ) Substitute values. = 0 Simplif. 4 Rise: Difference of -values Run: Difference of -values = 0 Slope is zero. (, ) Zero slope: line is horizontal. (, )

4 8. Find the slope of a line passing through (-, -) & (, -). EXAMPLE A Line with a Negative Slope Falls Find the slope of the line passing through (0, 0) and (, º). SOLUTION Let (, ) = (0, 0) and (, ) = (, º) º m = º = º º 0 º 0 = º Rise: Difference of -values Run: Difference of -values Substitute values. Simplif. = º Slope is negative. (0, 0) (, ) Negative slope: line falls from left to right. 9. Find the slope of a line passing through (-, ) & (, -).

5 0. Find the slope of a line passing through (, 0) & (0, ). Tell whether the line rises or falls from left to right. EXAMPLE 4 Slope of a Vertical Line is Undefined Find the slope of the line passing through (, 4) and (, ). SOLUTION Let (, ) = (, ) and (, ) = (, 4). º m = º = 4 º º Rise: Difference of -values Run: Difference of -values Substitute values. (, 4) (, ) = Division b 0 is undefined. 0 Because division b 0 is undefined, the epression 0 has no meaning. The slope of a vertical line is undefined. Undefined slope: line is vertical.. Find the slope of the line passing through (0, -) and (0, 4).. Find the slope of the line passing through (-, ) and (-, -).

6 EXAMPLE Given the Slope, Find a -Coordinate Find the value of so that the line passing through the points (º, ) and (4, ) has a slope of. SOLUTION Let (, ) = (º, ) and (, ) = (4, ). º m = º = º 4 º (º ) = º 6 Write formula for slope. Substitute values for m,,,, and. Simplif. 6 = 6 º Multipl each side b 6. 6 (, ) (4, ) 4 = º Simplif. = Add to each side.. Find the value of so that the line passing through the point (, ) and (, ) has a slope of -.

7 4. Find the value of so that the line passing through the point (, ) and (, 8) has a slope of.

8 REAL LIFE Parachuting PROBLEM SOLVING STRATEGY EXAMPLE 6 You are parachuting. At time t = 0 seconds, ou open our parachute at height h = 00 feet above the ground. At time t = seconds, ou are at height h = feet. a. What is our rate of change in height? b. About when will ou reach the ground? SOLUTION Slope as a Rate of Change Height (feet) Time (seconds) a. Use the formula for slope to find the rate of change. The change in time is º 0 = seconds. Subtract in the same order. The change in height is º 00 = º8 feet. VERBAL MODEL Rate of change = h 0 0 (0, 00) Change in height Change in time (, ) t LABELS Rate of change = m Change in height = º8 Change in time = (ft/sec) (ft) (sec) ALGEBRAIC = º 8 m MODEL Your rate of change is º ft/sec. The negative value indicates that ou are falling. b. Falling at a rate of ft/sec, find the time it will take ou to fall 00 feet. Time = Di stance 00 ft = 7 sec Rate ft/sec You will reach the ground about 7 seconds after opening our parachute. UNIT ANALYSIS Check that seconds are the units of the solution. ft sec ft = sec ft/ ft

9 . You are traveling b car and leave home at 8:00 A.M. B 8:4 A.M., ou are 6 miles from home. Find the average speed in miles per hour. 6. You are tping a paper. At : P.M. ou have tped 7 words. B :7 P.M. ou have tped 660 words. Find the average rate of change in words per minute. 7. Draw a ramp and label its rise and run. Eplain what is meant b the slope of the ramp. 8. Describe the error. Then calculate the correct slope.

10 9. Eplain what happens when the formula for slope is applied to a vertical line. 0. How can ou tell that the slope of the line through (, ) and (-, ) is negative without calculating?