c
2006-2012
F. Dos Reis
Such-That.com
Graphs and lines
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Cartesian Coordinates
Exercise 1.
3
2
1
B
A
1
O
2
3 4
• What are the coordinates of A? of B?
• Plot the point of C coordinates (1, −2).
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Graph of Ax + By = C
Definition: A linear equation in two variables is an equation that
can be written in the standard form
Ax + By = C
where A, B, and C are constant and A and B are not both 0. x and y
are variables.
Exercise 2. Given the linear equation
5x − 2y = 1.
• Find some solutions and plot them on the graph below.
• If x = 3, solve the equation for y. Plot the solution on the graph.
• If y = −1, solve the equation for x. Plot the solution on the graph.
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c
2006-2012
F. Dos Reis
Such-That.com
Theorem: The graph of any equation of the form
Ax + By = C
(A and B not both 0).
is a line.
Any line in a Cartesian coordinate system is the graph of an equation
of this form.
Remark: There exists several equations for the same line.
Exercise 3. Find a Cartesian equation of the vertical line and the horizontal line passing through
the point A(3, −4).
Exercise 4. Find a Cartesian equation of the line passing through the point A(0,1) and B(3,5).
Definition – Intercepts:
• If a line crosses the x axis at a point (a, 0), then a is an x intercept.
• If a line crosses the y axis at a point (0, b), then b is a y intercept.
Remark: To find the x intercepts, plug in 0 for y in the equation and solve for x.
To find the y intercept, plug in 0 for x in the equation and solve for y.
Exercise 5. Given the line 2x − 3y = 5.
• Find the x and y intercepts.
• Given x, solve the equation of the line for y.
• Graph the line in your calculator.
• Check the x intercept with your calculator.
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c
2006-2012
F. Dos Reis
Such-That.com
3
Slope of a Line
Definition – Slope of a line: If a line passes through two distinct
points A(xa , ya ) and B(xb , yb ).then its slope m is given by
m=
yb − ya
Vertical change (rise)
=
xb − xa
Horizontal change (run)
Remark: The slope measure the steepness of the line.
Exercise 6. What can we say about the slope of the lines below?
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Equation of Lines: Special Form
Definition – Slope-Intercept Form: The equation
y = mx + b
m = Slope,
b = y-intercept
is called the slope-intercept form of an equation line
Exercise 7. Find an equation of the line with slope m = −2 and y intercept 3 and graph it.
Exercise 8. Find a slope-intercept equation of the line defined by 3x + 2y = 7.
Definition – Point-slope Form: An equation of a line with slope m
that passes through A(xa , ya ) is
y − ya = m(x − xa )
which is called the point-slope form of an equation of a line.
3
c
2006-2012
F. Dos Reis
Such-That.com
Remark: Vertical lines are the only lines that don’t have a point-slope equation
Exercise 9. Find an equation of the line passing through the point A(3, 5) with slope m = −1.
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