Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.
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Transcript of Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.
![Page 1: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/1.jpg)
Pre-CalculusLesson 1: Algebra RevisitedFormulas, definitions, and methods from Algebra 1
![Page 2: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/2.jpg)
The distance d between any two points with coordinates and is given by the formula d = .
The Distance Formula
1 1( , )x y 2 2( , )x y
2 22 1 2 1( ) ( )x x y y
![Page 3: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/3.jpg)
Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)
![Page 4: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/4.jpg)
Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)
( 3 4)2 (2 1)2d =
![Page 5: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/5.jpg)
Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)
( 3 4)2 (2 1)2d =
( 7)2 (1)2 49 1d =
![Page 6: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/6.jpg)
Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)
( 3 4)2 (2 1)2d =
( 7)2 (1)2 49 1d =
50 or 5 2 or 7.07d =
![Page 7: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/7.jpg)
In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are .
Midpoint Formula
1 2 1 2,2 2
x x y y
1 1( , )x y 2 2( , )x y
![Page 8: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/8.jpg)
Applying the Midpoint FormulaFind the midpoint between (-2, 5) and (6, 4)
![Page 9: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/9.jpg)
Applying the Midpoint FormulaFind the midpoint between (-2, 5) and (6, 4)
2 62
,5 4
2
M =
![Page 10: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/10.jpg)
Applying the Midpoint FormulaFind the midpoint between (-2, 5) and (6, 4)
42
,92
= 2,
92
M =
2 62
,5 4
2
M =
![Page 11: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/11.jpg)
Domain and RangeDomain = the input of a functionRange = the output of a function
![Page 12: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/12.jpg)
Vertical Line Test: If a vertical line can connect two points
on the graph of a relation, then the relation is not a function.
Examples:
http://math.tutorvista.com/calculus/vertical-line-test.html
![Page 13: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/13.jpg)
Evaluating FunctionsTo evaluate a function, simply
replace (substitute) the function's variable with the indicated number or expression.
![Page 14: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/14.jpg)
Evaluating FunctionsTo evaluate a function, simply
replace (substitute) the function's variable with the indicated number or expression.
f(x) = 2x + 5, Find f(3).
![Page 15: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/15.jpg)
Evaluating FunctionsTo evaluate a function, simply
replace (substitute) the function's variable with the indicated number or expression.
f(x) = 2x + 5, Find f(3). f(3) = 2(3) + 5 = 11.
![Page 16: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/16.jpg)
Evaluating Functions
![Page 17: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/17.jpg)
Evaluating Functions
![Page 18: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/18.jpg)
Evaluating Functions
![Page 19: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/19.jpg)
Evaluating Functions
![Page 20: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/20.jpg)
Slope Formula:
12
12
xx
yym
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Find the slope of the line that goes through the points (-5, 3) and (2, 1).
Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS
![Page 22: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/22.jpg)
Find the slope of the line that goes through the points (-5, 3) and (2, 1).
m y2 y1
x2 x1
Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS
![Page 23: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/23.jpg)
Find the slope of the line that goes through the points (-5, 3) and (2, 1).
m y2 y1
x2 x1
1 3
2 ( 5)m
Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS
![Page 24: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/24.jpg)
Find the slope of the line that goes through the points (-5, 3) and (2, 1).
m y2 y1
x2 x1
1 3
2 ( 5)m
1 3
2 5m
Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS
![Page 25: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/25.jpg)
Find the slope of the line that goes through the points (-5, 3) and (2, 1).
m y2 y1
x2 x1
1 3
2 ( 5)m
1 3
2 5m
2
7m
Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS
![Page 26: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/26.jpg)
A few things to remember about circles:
A circle is a set of points equidistant from a central point.
A circle is named using its center point.
Radius=distance from center to edge of circle.
Diameter=2 x radius
![Page 27: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/27.jpg)
Equation of a CircleThe circle with center (0,0) and radius r is the graph of the equation
222 ryx
![Page 28: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/28.jpg)
Equation of a CircleThe circle with center (h,k) and radius r is the graph of the equation:
222 rkyhx
![Page 29: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/29.jpg)
Applying the Equation of a Circle Identify the coordinates of the
center and the length of the radius in the circle (x − 5)2 + (y + 2)2 = 4
Write the equation of a circle centered at (5,1) with a radius of 5
![Page 30: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/30.jpg)
Applying the Equation of a Circle Identify the coordinates of the
center and the length of the radius in the circle (x − 5)2 + (y + 2)2 = 4
(5, -2), r = 2Write the equation of a circle
centered at (5,1) with a radius of 5
![Page 31: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/31.jpg)
Applying the Equation of a Circle Identify the coordinates of the
center and the length of the radius in the circle (x − 5)2 + (y + 2)2 = 4
(5, -2), r = 2Write the equation of a circle
centered at (5,1) with a radius of 5 (x − 5)2 + (y – 1)2 = 25
![Page 32: Pre-Calculus Lesson 1: Algebra Revisited Formulas, definitions, and methods from Algebra 1.](https://reader035.fdocuments.us/reader035/viewer/2022081421/56649e6b5503460f94b69837/html5/thumbnails/32.jpg)
Extension ProblemWrite the equation of the circle whose diameter extends from the point (-2,1) to the point (6,-5).