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Graphs, Functions, and Models > Introduction to Graphing
Introduction to Graphing
• The Cartesian System
• Equations and Their Solutions
• Graphing Equations
• The Distance Formula and Midpoints of Segments
• Circles
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Graphs, Functions, and Models > Introduction to Graphing
The Cartesian System
• The Cartesian coordinate system is a 2-dimensional plane with a horizontal axis,
known as the x-axis, and a vertical axis, known as the y-axis.
• A Cartesian coordinate system specifies each point uniquely in a plane with a pair
of numerical coordinates, which are the signed distances from the point to the two
axes.
• Points are represented by an ordered pair (x, y), where the x-coordinate is the
point's distance from the y axis, and the y-coordinate is the distance from the xaxis.
2D Cartesian Coordinates
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Graphs, Functions, and Models > Introduction to Graphing
Equations and Their Solutions
• To solve an equation is to find what values (numbers, functions, sets, etc. ) fulfill a
condition stated in the form of an equation (two expressions related by equality).
• In mathematics, the graph of a function f is the collection of all ordered pairs (x,
f(x)), where f(x) is defined by some operation on x.
• Once a function has been graphed, solutions to any particular x or y value are
readily available by looking at the graph.
• To solve for an equation, you must simplify the equation by using algebraic
manipulations to get the variable by itself on one side (typically the left).
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Graphs, Functions, and Models > Introduction to Graphing
Graphing Equations
• Graphs are important tools to visualizing equations.
• To graph an equation, choose a value for either x or y, solve for the variable you
didn't choose to get a point, plot the point, and repeat until you have enough
points to draw the graph by connecting the points.
• Graphs can be closed, like a circle, or open, like a parabola.
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Graphs, Functions, and Models > Introduction to Graphing
The Distance Formula and Midpoints of Segments
• The Pythagorean Theorem is a very powerful theorem relating the lengths of the three sides of a
triangle. This theorem tells us that if c is the hypotenuse and a and b are the other two sides,
[Equation 1].
Equation 1
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Pythagorean Theorem
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Graphs, Functions, and Models > Introduction to Graphing
The Distance Formula and Midpoints of Segments
• Using the Pythagorean Theorem and two points (x1, y1) and (x2, y2), we can derive the distance
formula: [Equation 2].
Equation 2
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Pythagorean Theorem
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Graphs, Functions, and Models > Introduction to Graphing
The Distance Formula and Midpoints of Segments
• The midpoint of a line segment given by two points (x1, y1) and (x2, y2) is [Equation 3].
Equation 3
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Pythagorean Theorem
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Graphs, Functions, and Models > Introduction to Graphing
Circles
• A circle is defined as the set of points that are a fixed distance from a center
point.
• The distance formula can be extended directly to the definition of a circle by
noting the radius is simply the distance between the center of a circle and the
edge.
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Graphs, Functions, and Models > Introduction to Graphing
• The general equation for a circle, centered at (a, b), with radius r is the set of all points (x, y) such
that [Equation 4].
Equation 4
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Graphs, Functions, and Models > Introduction to Graphing
• Pi is the ratio of a circle's circumference to its diameter.
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Appendix
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Graphs, Functions, and Models
Key terms
• circle A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from
another point.
• distance The amount of space between two points, measured along a straight line
• equation An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign.
E.g. x=5.
• expression An arrangement of symbols denoting values, operations performed on them, and grouping symbols. E.g. (2x+4)/2
• graph A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or
numbers.
• graph A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or
numbers.
• midpoint A point which divides a line segment into two lines of equal length
• ordered pair A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system.
Notation: (x, y).
• point An entity that has a location in space or on a plane, but has no extent
• Pythagorean Theorem The Pythagorean Theorem, originally put forth by Pythagoras, states that the square of the hypotenuse
is equal to the sum of the squares of the other two sides in a right triangle.
• quadrant the four regions of the Cartesian plane bounded by the x-axis and y-axis
• radius A line segment between any point on the circumference of a circle and its center/centre.
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Graphs, Functions, and Models
• x-axis The axis on a graph that is usually drawn left to right, with values increasing to the right.
• y-axis The axis on a graph that is usually drawn from bottom to top, with values increasing as they go up.
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Graphs, Functions, and Models
Interactive Graph: A Circle
This is a circle based on the equation . Try changing the size of the circle by playing with the equation.
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Graphs, Functions, and Models
Distance Formula
The distance formula between two points, (x1, y1) and (x2, y2), shown as the hypotenuse of a right triangle
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Graphs, Functions, and Models
Interactive Graph: Graph of a Cubic Polynomial Function
This is the graph of the function , a cubic polynomial. Using what you now know about Cartesian planes, examine where the graph crosses the x-axis
and the y-axis. There's a reason why it looks the way it does.
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Graphs, Functions, and Models
Interactive Graph: Parabola Graph
This is a graph of a parabola given by the equation . Take note of where the ordered pairs you plotted are. What other points fit on the parabola?
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Graphs, Functions, and Models
Interactive Graph: A Circle
The equation x2+y2=100 results in a circle with radius 10 centered on the origin. Notice what happens to the radius when you change the different
variables in the equation.
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Graphs, Functions, and Models
2D Cartesian Coordinates
The four quadrants of a Cartesian coordinate system. The arrows on the axes indicate that they extend forever in their respective directions (i.e.
infinitely).
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Graphs, Functions, and Models
Midpoint of a Line Segment
The equation for a midpoint of a line segment with end points (x1, y1) and (x2, y2)
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Wikipedia. "Midpoint." GNU FDL http://en.wikipedia.org/wiki/File:Midpoint.svg View on Boundless.com
Graphs, Functions, and Models
Pythagorean Theorem
The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
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Wikimedia. "Pythagorean theorem abc." GNU FDL http://commons.wikimedia.org/wiki/File:Pythagorean_theorem_abc.svg View on Boundless.com
Graphs, Functions, and Models
Which of the following is NOT true about the Cartesian coordinate
system?
A) The point where the axes meet is called the origin.
B) The horizontal axis is called the y-axis.
C) The axes divide the plane into four quadrants.
D) It is named after the 17th-century French mathematician René
Descartes.
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Graphs, Functions, and Models
Which of the following is NOT true about the Cartesian coordinate
system?
A) The point where the axes meet is called the origin.
B) The horizontal axis is called the y-axis.
C) The axes divide the plane into four quadrants.
D) It is named after the 17th-century French mathematician René
Descartes.
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Graphs, Functions, and Models
In what quadrants do the following points lie, respectively: (1,-2);
(-2,-4); (3,4); (-7,8)?
A) IV, II, I, III
B) IV, III, I, II
C) I, II, III, IV
D) II, IV, III, I
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Graphs, Functions, and Models
In what quadrants do the following points lie, respectively: (1,-2);
(-2,-4); (3,4); (-7,8)?
A) IV, II, I, III
B) IV, III, I, II
C) I, II, III, IV
D) II, IV, III, I
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Graphs, Functions, and Models
An assertion that two expressions are equal, expressed by writing
the two expressions separated by an equals sign. E.g. x=5.
A) rational number
B) equation
C) associative
D) absolute value
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Graphs, Functions, and Models
An assertion that two expressions are equal, expressed by writing
the two expressions separated by an equals sign. E.g. x=5.
A) rational number
B) equation
C) associative
D) absolute value
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Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com/
Graphs, Functions, and Models
Which of the following functions does NOT have a solution for
every value of x?
A) f(x) = 2x + 1
B) f(x) = 7 - x2
C) f(x) = 3
D) f(x) = 3, if x=2
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Graphs, Functions, and Models
Which of the following functions does NOT have a solution for
every value of x?
A) f(x) = 2x + 1
B) f(x) = 7 - x2
C) f(x) = 3
D) f(x) = 3, if x=2
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Graphs, Functions, and Models
Given the function f(x) = x3 - 9x, what is the value of f(x) when x =
2?
A) 26
B) -8
C) -10
D) No solution
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Graphs, Functions, and Models
Given the function f(x) = x3 - 9x, what is the value of f(x) when x =
2?
A) 26
B) -8
C) -10
D) No solution
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Graphs, Functions, and Models
How many points are needed to draw the graph of a straight line?
A) 1
B) 2
C) 3
D) 4
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Graphs, Functions, and Models
How many points are needed to draw the graph of a straight line?
A) 1
B) 2
C) 3
D) 4
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Graphs, Functions, and Models
Which of the following ordered pairs does NOT fall on the line 5x+3y=15?
A) (0,5)
B) (-3,0)
C) (1,7)
D) (-2, 5/3)
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Graphs, Functions, and Models
Which of the following ordered pairs does NOT fall on the line 5x+3y=15?
A) (0,5)
B) (-3,0)
C) (1,7)
D) (-2, 5/3)
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Graphs, Functions, and Models
What is the midpoint of (-1,2) and (3,-6)?
A) (-1,-2)
B) (-1,2)
C) (1,-2)
D) (1,2)
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Graphs, Functions, and Models
What is the midpoint of (-1,2) and (3,-6)?
A) (-1,-2)
B) (-1,2)
C) (1,-2)
D) (1,2)
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Graphs, Functions, and Models
What is the distance between the points (-2,1) and (1,_______)?
A) 5
B) 3
C) 4
D) 25
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Graphs, Functions, and Models
What is the distance between the points (-2,1) and (1,_______)?
A) 5
B) 3
C) 4
D) 25
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Graphs, Functions, and Models
What is the distance between (8,-2) and (3,9), rounded to the
nearest hundredth?
A) 8.60
B) 12.08
C) 8.49
D) 11.65
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Graphs, Functions, and Models
What is the distance between (8,-2) and (3,9), rounded to the
nearest hundredth?
A) 8.60
B) 12.08
C) 8.49
D) 11.65
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Graphs, Functions, and Models
Which of the following is NOT true about circles?
A) r=2d
B) C=πd
C) A=πr2
D) r2=(x-a)2+(y-b)2
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Graphs, Functions, and Models
Which of the following is NOT true about circles?
A) r=2d
B) C=πd
C) A=πr2
D) r2=(x-a)2+(y-b)2
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Graphs, Functions, and Models
What is the circumference of a circle with a radius of 4 in (to the
nearest hundredth)?
A) 12.56 in
B) 25.12 in
C) 6.28 in
D) 50.24 in
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Graphs, Functions, and Models
What is the circumference of a circle with a radius of 4 in (to the
nearest hundredth)?
A) 12.56 in
B) 25.12 in
C) 6.28 in
D) 50.24 in
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Graphs, Functions, and Models
What is the area of a circle with a diameter of 6 cm (to the nearest
hundredth)?
A) 113.04 cm2
B) 9.42 cm2
C) 28.26 cm2
D) 18.84 cm2
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Graphs, Functions, and Models
What is the area of a circle with a diameter of 6 cm (to the nearest
hundredth)?
A) 113.04 cm2
B) 9.42 cm2
C) 28.26 cm2
D) 18.84 cm2
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Graphs, Functions, and Models
Attribution
• Wikibooks. "Algebra/Equations." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Equations
• Wikibooks. "Algebra/Function Graphing." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Function_Graphing
• Wikipedia. "point." CC BY-SA 3.0 http://en.wikipedia.org/wiki/point
• Wiktionary. "graph." CC BY-SA 3.0 http://en.wiktionary.org/wiki/graph
• Wikipedia. "Cartesian coordinate system." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cartesian_coordinate_system
• Wikibooks. "Algebra/The Coordinate (Cartesian) Plane." CC BY-SA 3.0
http://en.wikibooks.org/wiki/Algebra/The_Coordinate_(Cartesian)_Plane
• Wikibooks. "Intermediate Algebra/The Coordinate (Cartesian) Plane." CC BY-SA 3.0
http://en.wikibooks.org/wiki/Intermediate_Algebra/The_Coordinate_(Cartesian)_Plane
• Wiktionary. "quadrant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/quadrant
• Wiktionary. "y-axis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/y-axis
• Wiktionary. "x-axis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/x-axis
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/ordered-pair
• Wikibooks. "Algebra/Circle." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Circle
• Wiktionary. "radius." CC BY-SA 3.0 http://en.wiktionary.org/wiki/radius
• Wiktionary. "circle." CC BY-SA 3.0 http://en.wiktionary.org/wiki/circle
• Wikipedia. "Circle." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Circle
• Wikibooks. "Algebra/The Pythagorean Theorem." CC BY-SA 3.0
http://en.wikibooks.org/wiki/Algebra/The_Pythagorean_Theorem
• Wikipedia. "Midpoint." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Midpoint
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Graphs, Functions, and Models
• Wiktionary. "distance." CC BY-SA 3.0 http://en.wiktionary.org/wiki/distance
• Wiktionary. "midpoint." CC BY-SA 3.0 http://en.wiktionary.org/wiki/midpoint
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/pythagorean-theorem
• Wikipedia. "Distance formula." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Distance_formula#Geometry
• Wikibooks. "Algebra/Function Graphing." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Function_Graphing
• Wikibooks. "Algebra/Equations." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Equations
• Wikibooks. "Algebra/Solving Equations." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Solving_Equations
• Wikipedia. "Equation solving." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Equation_solving
• Wikipedia. "Graph of a function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Graph_of_a_function
• Wikipedia. "Equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Equations
• Wiktionary. "graph." CC BY-SA 3.0 http://en.wiktionary.org/wiki/graph
• Wiktionary. "equation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/equation
• Wiktionary. "expression." CC BY-SA 3.0 http://en.wiktionary.org/wiki/expression
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