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Math
A |
Solving
a Linear-Quadratic System Graphically |
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Solving
a linear-quadratic system of equations
graphically
involves following a series of steps. |
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The
easiest way to illustrate this is to follow the steps through in a
specific example-------so, here goes.
| Example
1 |
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| 1. |
Solve
the following system of equations graphically:
y = x2
- 4x - 2
y = x -
2 |
First
we will graph the quadratic equation:
y = x2-
4x- 2 |
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We
recognize this as the graph of a parabola, since it fits the
form:
y = ax2
+ bx
+ c
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Since
no values are specified for x, we will first find the equation of
the axis of symmetry. |
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To
do this we use the formula
x = -b/2a
In this example, a =1, b=-4,
and c = -2. |
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Substituting
we get:
x = -(-4)/2(1)
x = 4/2
x = 2
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| Now
we know that the turning point, which lies on the axis of symmetry, has an
x-coordinate of 2. |
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This will be our
middle
value for x. We will choose 3 values less than 2 and 3 values
greater than 2.
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| Substitute
each value of x in the quadratic equation to find the corresponding values
for y. |
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For
example, substituting -1 for x we get
y = (-1)2 - 4(-1) - 2
y = 1 + 4 - 2
y = 3
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we do this for each value of x we get the table to the right.
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x |
y |
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-1
0
1
2
3
4
5 |
3
-2
-5
-6
-5
-2
3 |
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Next we graph
the points from the table to get the graph of the parabola to the right.
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O.K. |
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Done
with
the first
step ! |
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we'll graph the linear equation y = x - 2 on the same
set of axes. |
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To
do that we need to know the slope and the y-intercept.
Standard form for a line is:
y = mx + b
where m
is the slope, and b is the y-intercept. |
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Our
equation is y = x - 2, so
m = 1, and b = -2 |
| Draw
the graph starting at -2
on the y-axis. Use slope (which is rise over run) to find other points by going up
1 and
to the right
1, or down
1 and
to the left
1. |
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y = x2
- 4x - 2
y
= x - 2
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| The
last step is to find the point(s) where the two grpahs
intersect. This is the solution set of the system of
equations. |
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Our
graphs intersect at 2 points whose coordinates are:
(0,-2) and (5,3)
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So
the solution set is:
{(0,-2),(5,3)} |
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How to use
your
TI-83+ graphing calculator with
quadratic equations.
Click calculator. |
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