7.
Parameters
In
this section we apply our understanding of transformations to study the
influence of parameters on the graphs of known functions.
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Let’s
investigate the influence of the parameters a, b and c on
the graph of y
= ax2
+ bx + c as a, b and c change.
Open
the Applet below. Click on each of the sliders to see how
each of a, b and c influence the graph. Then in each case
describe how the graph changes as the parameter changes. Can
you explain the influence?
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Let’s
continue our investigation of the influence of the parameters
a, b and c on the graph of
y
= ax2
+ bx + c as a, b and c change.
Click
on each of the buttons below to see an animation for each
of a, b and c. Then in each case describe how the graph changes
as the parameter changes. Can you explain the influence?
See the discussion below.
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Open
the Vertex locus applet below to investigate the locus
of
the vertex of y
= ax2
+ bx
+ c as a, b and c change. To
investigate, you must vary the parameters a, b and c systematically.
For example:
-
Investigate c:
The applet starts with a = 1, b = -8 and c = 4. Leave a
and b and vary the value of c by clicking on the c-slider.
How will you describe the locus of the vertex? What is the
equation of the locus of the vertex? Check your equation
by typing the equation into the purple input box and pressing
ENTER. (For example y
= 2*x+3.
You must use small x
and y,
use * for multiplication and ^ for exponentiation.) Click
“init” to reset the applet. Now change a to
a = -4, click “clear” to clear the traces, and
vary c again. Describe the shape of the locus, find its
equation and check the equation. Can you generalize? Can
you give the general equation of the locus of the vertex
of y
= ax2
+ bx
+ c if a and b are constant and c varies? Investigate other
values if necessary.
- Now
investigate b: Click “init” to reset the
applet. Leave a and c and vary the value of b by clicking
on the b-slider. How will you describe the locus of the
vertex? What is the equation of the locus of the vertex?
Check your equation by typing the equation into the purple
input box and pressing ENTER. Click “init” to
reset the applet. Now change a to a = -1, click “clear”
to clear the traces, and vary b again. Describe the shape
of the locus, find its equation and check the equation.
Try other values like a = 2. Can you generalize? Can you
give the general equation of the locus of the vertex of
y
= ax2
+ bx
+ c if a and c are constant and b varies?
- Now
investigate a:
Click “init” to reset the applet. Leave b and
c and vary the value of a by clicking on the a-slider. Continue
… Now change b to b = -4 and vary a again. Describe
the shape of the locus, find its equation and check the
equation. Try other values like b = 6. Can you generalize?
Can you give the general equation of the locus of the vertex
of y
= ax2
+ bx
+ c if b and c are constant and a varies?
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Open the Standard form applet for the parabola with
equation y
= a(x - p)2 + q:
Click
the sliders and make a careful analysis of the influence of
a, p and q …
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