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- Precalculus
- Linear functions
- Graphs of linear functions
Key concepts
- Linear functions may be graphed by plotting points or by using the
y -intercept and slope. See
[link] and
[link] .
- Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See
[link] .
- The
y -intercept and slope of a line may be used to write the equation of a line.
- The
x -intercept is the point at which the graph of a linear function crosses the
x -axis. See
[link] and
[link] .
- Horizontal lines are written in the form,
See
[link] .
- Vertical lines are written in the form,
See
[link] .
- Parallel lines have the same slope.
- Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See
[link] .
- A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the
x - and
y -values of the given point into the equation,
and using the
that results. Similarly, the point-slope form of an equation can also be used. See
[link]
.
- A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See
[link] and
[link] .
- A system of linear equations may be solved setting the two equations equal to one another and solving for
The
y -value may be found by evaluating either one of the original equations using this
x -value.
- A system of linear equations may also be solved by finding the point of intersection on a graph. See
[link] and
[link] .
Section exercises
Verbal
If the graphs of two linear functions are parallel, describe the relationship between the slopes and the
y -intercepts.
The slopes are equal;
y -intercepts are not equal.
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If a horizontal line has the equation
and a vertical line has the equation
what is the point of intersection? Explain why what you found is the point of intersection.
The point of intersection is
This is because for the horizontal line, all of the
coordinates are
and for the vertical line, all of the
coordinates are
The point of intersection will have these two characteristics.
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Explain how to find a line perpendicular to a linear function that passes through a given point.
First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation
and solve for
Then write the equation of the line in the form
by substituting in
and
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Algebraic
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
Questions & Answers
Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
can you send the book attached ?
Ariel
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2
find mu of x for y
U(x,y) = (xÃy)1/2
find mu of x for y
Desalegn
this is the study of how the society manages it's scarce resources
Belonwu
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
difference between firm and industry
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from
Qd 120 3P
to
Qd 90 3P
and the supply function has shifted from
QS
20 2P
to
QS
10 2P .
a) Find the effect of this change on price and quantity.
b) Which of the changes in demand and supply is higher?
explain standard reason why economic is a science
factors influencing supply
scares
means__________________ends
resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Why qualify 28 supplies
Milan
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
identify a demand and a supply curve
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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