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Transcript of Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. ...
![Page 1: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/1.jpg)
Equation of a LineThm. A line has the equation y = mx + b,
where m = slope and b = y-intercept.
This is called Slope-Intercept Form
Ex. Find the slope and y-intercept:a) y = 3x + 1
b) 2x + 3y = 1
![Page 2: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/2.jpg)
Ex. Graph:a) 2 1y x
4
2
-2
-4
-5 5
![Page 3: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/3.jpg)
Ex. Graph:b) 2y
4
2
-2
-4
-5 5
![Page 4: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/4.jpg)
Ex. Graph:c) 2x y
4
2
-2
-4
-5 5
![Page 5: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/5.jpg)
SlopeThm. The slope between (x1,y1) and (x2,y2) is
Ex. Find the slope between (-2,0) and (3,1).
2 1
2 1
y ym
x x
![Page 6: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/6.jpg)
• Rising line has positive slope:
• Falling line has negative slope:
• Horizontal line as slope of 0:
• Vertical line has undefined slope:
A vertical line has an equation like x = 3, and can’t be written as y = mx + b
25 , 4
378,
0 05 3,
820 0,
![Page 7: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/7.jpg)
Thm. Parallel lines have the same slope.
Thm. Perpendicular lines have slopes that are negative reciprocals.
Ex. Are the lines parallel, perpendicular, or neither? (3,-1) to (-3,1) and (0,3) to (-1,0)
![Page 8: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/8.jpg)
Thm. A line with m = slope that passes through the point (x1,y1) has the equation
y – y1 = m(x – x1)
This is called Point-Slope Form
Ex. Write the equation in Slope-Intercept Form:a) slope = 3, contains (1,-2)
![Page 9: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/9.jpg)
Ex. Write the equation in Slope-Intercept Form:b) contains (2,5) and (4,-1)
![Page 10: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/10.jpg)
Ex. Find equation of the lines that pass through (2,-1) are:a) parallel to, andb) perpendicular to, the line 2x – 3y = 5
![Page 11: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/11.jpg)
Ex. The maximum slope of a wheelchair ramp is . A business is installing a ramp that rises 22 in. over a horizontal distance of 24 ft. Is the ramp steeper than required?
112
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Ex. An appliance company determines that the total cost, in dollars, of producing x blenders is
C = 25x + 3500Explain the significance of the slope and y-intercept.
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Ex. A college purchases exercise equipment worth $12,000. After 8 years, the equipment is determined to have a worth of $2000. Express this relationship as a linear equation. How many years will pass before the equipment is worthless?
![Page 14: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/14.jpg)
Practice Problems
Section 2.1
Problems 9, 21, 41, 51, 69, 107, 109
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Functions
Def. (formal) A function f from set A to set B is a relation that assigns to each element x in set A exactly on element y in set B.
Def. (informal) A set of ordered pairs is a function if no two points have the same x-coordinate
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Set A(the x’s)
The input
Domain
Set B(the y’s)
The output
Range
x is called the independent variabley is called the dependent variable
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![Page 18: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/18.jpg)
Ex. Determine whether the relation is a function:
a)
b)
40-375y
3-1230x
2
-2
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Ex. Determine whether the relation is a function:
c) x is the number of representatives from a state, y is the number of senators from the same state
d) x is the time spent at a parking meter, y is the cost to park
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Ex. Determine whether the relation is a function:
e) x2 + y = 1
f) x + y2 = 1
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Rather than writingy = 7x + 2
we can express a function asf (x) = 7x + 2
This is called Function Notation
![Page 22: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/22.jpg)
Ex. Let g(x) = -x2 + 4x + 1, find
a) g(2)
b) g(t)
c) g(x + 2)
![Page 23: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/23.jpg)
The next example is called a piecewise function because the equation depends on what we are plugging in.
Ex. Let , find f (-1),
f (0), and f (1).
2 1 0
1 0
x xf x
x x
![Page 24: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/24.jpg)
Finding the domain means determining all possible x’s that can be put into the function
Ex. Find the domain of the functionf : {(-3,0), (-1,4), (0,2), (2,2), (4,-1)}
![Page 25: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/25.jpg)
Often, finding the domain means finding the x’s that can’t be used in the function
Ex. Find the domain of the function
a)
b) Volume of a sphere:
1
5g x
x
343V r
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Ex. Find the domain of the functionc) 24h x x
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Ex. You’re making a can with a height that is 4 times as long as the radius. Express the volume of the can as a function of height.
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Ex. When a baseball is hit, the height of a baseball is given by the function f (x) = -0.0032x2 + x + 3, where x is distance travelled (in ft) and f (x) is height (in ft). Will the baseball clear a 10-foot fence that is 300 ft from home plate?
![Page 29: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/29.jpg)
Ex. For f (x) = x2 – 4x + 7, find f x h f x
h
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Practice Problems
Section 2.2
Problems 9, 15, 29, 35, 59, 79, 87, 93
![Page 31: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/31.jpg)
Graph of a Function
Ex. Using the graph, find:
a) domain
b) range
c) f (-1), f (1), and f (2)
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The graph of a function will pass the vertical line test – all vertical lines will pass through the graph at most once.
Ex. Determine if this is the graph of a functiona)
4
2
b) 4
2
-5
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Def. The zeroes of a function f are the x-values for which f (x) = 0.
Ex. Find the zeroes of the functiona) f (x) = 3x2 + x – 10
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Ex. Find the zeroes of the functionb)
c)
This is where the graph crosses the x-axis.
2 3
5
th t
t
210g x x
![Page 35: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/35.jpg)
Let discuss increasing, decreasing, relative minimum, and relative maximum
4
3
2
1
2 4 6 8 10
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Ex. Use a calculator to approximate the relative minimum of the function f (x) = 3x2 – 4x – 2.
![Page 37: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/37.jpg)
Earlier, we worked with slope as the rate of change of a line
If the graph is nonlinear, we still want to talk about rate of change, but this slope is different at every point.
![Page 38: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/38.jpg)
We can discuss the average rate of change between two points.
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-0.2
0.5 1 1.5 2 2.5 3 3.5 4
(x1,y1)
(x2,y2)
![Page 39: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/39.jpg)
The points can be connected using a secant line
(x1,y1)
(x2,y2)
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-0.2
0.5 1 1.5 2 2.5 3 3.5 4
![Page 40: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/40.jpg)
The average rate of change is the slope between the points
(x1,y1)
(x2,y2)
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-0.2
0.5 1 1.5 2 2.5 3 3.5 4
2 1
2 1
y ym
x x
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Ex. Find the average rate of change of f (x) = x3 – 3x from x1 = -2 to x2 = 0.
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Ex. The distance s (in feet) a moving car has traveled is given by the function , where t is time (in seconds). Find the average speed from t1 = 4 to t2 = 9.
3220s t t
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Def. A function f (x) is even if f (-x) = f (x).
The graph will have y-axis symmetry
4 22y x x
1.5
1
0.5
-0.5
-1
-1 1
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Def. A function f (x) is odd if f (-x) = - f (x).
The graph will have origin symmetry1.5
1
0.5
-0.5
-1
-1 1
3 2y x x
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Ex. Determine if the function is even, odd, or neither:
a) g(x) = 3x3 – 2x
b) h(x) = x2 + 1
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Ex. Determine if the function is even, odd, or neither:
c) f (x) = x3 – 4x + 8
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Practice Problems
Section 2.3
Problems 3, 10, 15, 33, 54, 63, 71, 89, 93
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Parents FunctionsWe are going to talk about some basic
functions, and next class we will expand upon them.
Earlier, we saw that a function f (x) = ax + b is linear
The domain of a linear function is all real numbers, and the range is all real numbers
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The constant function is f (x) = c
The graph is a horizontal line
4
2
-2
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The identify function is f (x) = x
2
-2
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The squaring function is f (x) = x2
The domain is all real numbersThe range is all nonnegative numbersThe graph is even and has y-axis symmetry
4
2
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The cubic function is f (x) = x3
The domain is all real numbersThe range is all real numbersThe graph is odd and has origin symmetry
2
-2
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The reciprocal function is
The domain is all nonzero numbersThe range is all nonzero numbersThe graph is odd and has origin symmetry
1f x
x
3
2
1
-1
-2
-2 2
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Ex. Sketch a graph of 2 3 1
4 1
x xf x
x x
6
4
2
-2
-4
-6
-5 5
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Def. The greatest integer function, , is defined as
x
the greatest integer
less than or equal tof x x
x
1.5
85 5 3.7
![Page 56: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/56.jpg)
The graph looks like this:
This type of function is called a step function
![Page 57: Equation of a Line Thm. A line has the equation y = mx + b, where m = slope and b = y-intercept. This is called Slope-Intercept Form Ex. Find the slope.](https://reader037.fdocuments.in/reader037/viewer/2022110401/56649dde5503460f94ad7af5/html5/thumbnails/57.jpg)
Practice Problems
Section 2.4
Problems 29, 43