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Section 6.2-6.3: Rectangular Coordinates
Practice HW from Mathematical Excursions Textbook (not to hand in) p. 350 # 1, 7-35 odd, 43-51 odd p. 360 # 1-21 odd
Linear Functions
A linear function is a function of the form y^ ^ f (^ x ) mx b where m = slope and b represents the y -coordinate of the y -intercept (^0 , b^ ). Note: The graph of a linear function is a straight line. To graph a line, we need at least two points. A quick way to sketch a graph of a linear function is to find its intercepts. Definition: For a linear function y^ ^ f (^ x ) mx b , we define the intercepts as follows:
- y - intercept – The point where the graph of the function crosses the y -axis. To find the y - intercept, set x = 0 and solve for y. This will give the point (0, b ).
- x - intercept – The point where the graph of the function crosses the x -axis. To find the x - intercept, set y = 0 and solve for x. This will give the point (^ ,^0 ) a b (^). Example 1: Find the x - and y - intercepts of the graph of the equation f^ (^ x )^2 x ^4.
Example 2: Find the x - and y - intercepts of the graph of the equation 4 x^ ^3 y ^8. Solution: █
Facts about Lines
- The slope intercept equation of a line is given by y f ( x ) mx b where m = slope and b represents the y -coordinate of the y -intercept (^0 , b ) .
- If m > 0, then the line goes up from left to right. If m < 0, then the line goes down from left to right.
- Suppose we are given the points two points ( x 1 (^) , y 1 ) and ( x 2 (^) , y 2 ) on the following line. We define the slope through these points as follows: 2 1 2 1 run rise change in change in x x y y x y m
- If m = 0, then y^ ^ f (^ x )^0 x b or f^ (^ x ) b. This gives a constant function. x y
x
y
Writing the Equation of a Line
To write the equation of any line, we need the slope and at least one point. Using y mx b we substitute the value of value of m and the coordinates of the point in for x and y and solve for b. Example 5: Find the equation of the line that passes through the point (2, 1) and has slope of -3. Solution: █
Example 6: Find the equation of the line through the points (^1 ,^2 )and (^5 ,^8 ). Sketch the graph. Solution: █
Fact: For the equation y^ ^ f (^ x ) mx b , the slope m means that when x increases by one unit, y increases (if m^ ^0 )or decreases (if m^ ^0 )by m units. Example 8: During a brisk walk, a person burns about 3.8 calories per minute. If a person has burned 191 calories in 50 minutes, determine a linear function that models the number of calories burned after t minutes. Use it to determine the number of calories burnt after 1 hour (60 minutes). Solution: Here, we want a linear function that represents the number of calories burned as a function of the number of minutes t. If we let C = the number of calories burned and t = the number of minutes, then the linear function that models this problem is C mt b To complete the equation of this linear function, we need the slope and a point that satisfies this function. Since a person burns 3.8 calories per minute, when we increase the time by 1 minute, the calories burnt increases by 3.8. Thus, the slope is m 3. 8. Hence, substituting this value gives C 3. 8 t b To find b , we need a point that satisfies this equation. Points on this equation are of the form ( t , C ). If a person has burned 191 calories in 50 minutes, this says t = 50 when C = 191, thus giving the point (50, 191). Substituting these values gives 191 3. 8 ( 50 ) b or 1 191 190 191 190 191 3. 8 ( 50 ) b b b b Thus, the linear function describing the number of calories burnt is C 3. 8 t 1 To find the number calories burnt after 60 minutes, we simply substitute t = 60 into this equation.
- 8 ( 60 ) 1 229 calories 60 Burntafter 60 minutes Numberof Calories C t █ Example 9: As a weather balloon rises in altitude from sea level, the temperature decreases at a fairly constant rate. If the temperature is 59 o^ Fat sea level and (^55). 5 oF
at 1000 ft, find a linear function that relates the altitude to the temperature. What is the temperature at an altitude of 24000 ft? █
