Artifact 1: Triangle and its medians
Part 1: Exploration
Statement of mathematical Exploration
For this artifact I will explore in TI-Nspire graphs of power functions. The
exploration is broken into two parts. The First part examines the effects of parameter
changes on the graph of a power function. The second explores families of power functions
with a given property.
My Mathematical Exploration
Exploration 1: Slider Exploration
For the initial investigation I used TI-NSpire to examine the graph of the parabola
𝑦 = 𝑎 ∗ 𝑥 𝑏 + 𝑐 for different values of a, b and c, where a ,b and c are any rational number. I
graphed the function and created sliders for a, b and c to explore how they affect the graph.
At first I started by changing the value of b. First I set b=0, leaving a=1 and c=0. This created
a straight, horizontal line. I then changed b=1 and this created a straight line again but will
a positive slope. When b=2 the graph displayed a upward facing parabola. These all make
sense because when I change b I am changing the degree of the equation. When b=0 the
equation will be a(1)+c and since a and c are both constants the graph will be a horizontal
line. Similarly when b=1 the equation will be a*x1+c, which is a linear equation.
As I continued this pattern I noticed that when b is even the end behavior of the
graph is positive as x goes to both infinity and negative infinity. When b is odd the end
behavior is negative when x goes to negative infinity and positive when x goes to positive
infinity.
Next I explored what happens when b is negative. When b is negative the graph has
an asymptote at x=0. When b is negative and odd, as x approaches 0 the graph approaches
negative infinity from the left and positive infinity from the right. When b is negative and
even, as x approaches 0 the graph goes to positive infinity from both the left and the right.
Next I explored what would happen to these graphs if I changed the parameter, a.
First I set slider b=0 to see how a would affect this graph. The graph of the horizontal line
created when b=0 and c=0, shifted the graph up and down the value of a on the y-axis.
Next I let b=1 and manipulate the parameter a. I discovered that when a gets farther
and farther from 0 the slope of the linear line becomes more and more steep. Conversely,
the closer a gets to 0 the more flat the slope of the line becomes.
When a=0 the line becomes a horizontal line at the value of c. This makes sense because a is
the coefficient of the x term and the equation will simply become y=c when a=0.
Similar to when a is positive, when a is negative the slope becomes steeper as the
value of a approaches negative infinity and flatter as the value a approaches 0. Except when
a is negative the slope of the line is also negative.
Still focusing on parameter a, I set parameter b=2. As I moved the value of a around I
observed that the parabola opens up when a is positive and opens down when a is negative.
The larger the absolute value of a, the skinnier the parabola will be and the smaller the
absolute value of a, the fatter the parabola will be.
When looking at a and b=3, similar things happen to the graph. I can conclude that when a
is negative the original graph is reflected over the horizontal line, y=c. I found that function
reflects over y=c and not the x-axis since the turns in the graph below are not below the xaxis as they should be if the function was reflected over the x-axis.
Also the closer the value of a is to 0 the skinnier the graph will be and the graph will
be fatter the closer to value of a is to 0.
Next I examined the parameter, c. Initially I let a=1 and b=1. As I moved the slider
for c the graph of the function shifter up and down the value of c. The slope and everything
else about the graph remained the same.
Then I set a=1 and b=2. Similarly, the parabola shifted up and down as I changed the
value of c. Again the same pattern followed when b=3, 4, 5, etc.
Next I set b=0 and a=1 and looked at how parameter c affected the graph. The graph of this
horizontal line is shifted up or down by c+1 when the c is manipulated.
When I change a=2 and manipulate c, the horizontal line is shifted up or down by
c+2. I can see how this works because when b=0 the equation of the graph is y=a(1)+c.
Exploration 2: Simultaneous Exploration
a.) Case 1: b, c are constant a varies: f(x) = a*xb+c | a = {-4, -3, -2, -1, 0, 1, 2, 3, etc}. Where b
and c are any constant number such that b and c are not equal.
First I looked at the affect “a” had on the graph when b=0 and c=1. When I graphed
each value of a simultaneously I got that the lines were horizontal lines where y=a+c.
From this graph I can see that “a” shifts the graph up and down, when b=0 and c=1. Next I
set b=1 and left c=1 and simultaneously graphed different values of a.
I noticed that the value of a changes the slope of the linear line formed when b=1 and c=1.
When I change b to 2 and keep c=1, it is hard to tell exactly what is going on with the
different equations on the same graph.
To solve this problem, I changed the color of some subsets of the values of a used. First I
looked at the positive values of a (orange) versus the negative values of a (brown). I saw
that the positive values made the parabola open up and the negative values made it open
down.
Then I looked at the odd versus even values of a. There was no distinct pattern I noticed
depending on a being even (red) or odd (green). I did notice however that the larger the
value of |a| the fatter the parabola would be. I also noticed that the vertexes of these
parabolas are the same and are located at the value of c on the y-axis.
Next I changed b=3 to see if the value of a affected the graph any differently than above.
The graphs did not look the same as above but when a is positive(orange) versus when a is
negative(brown) that result is similar to when b=2. The graphs reflect over y=c when a is
negative.
b.) Case 2: a, c are constant a varies: f(x) = a*xb+c | b = {-4, -3, -2, -1, 0, 1, 2, 3, etc}. Where a
and c are any constant number such that a and c are not equal.
First I looked at the affect “b” had on the graph when a=1 and c=1. Once again it is hard to
see what is going on when all of the lines are graphed on the same coordinate plane.
So I looked at the positive values of b (orange) versus the negative values of b (brown). I
noticed only that positive values for b cross the y-axis, meaning the negative values of b
have a vertical asymptote at x=0.
Next, I color coated the different equations so that I could see what is happening.
b=-3(blue), -2(red), -1(black), 0(pink), 1(green), 2(orange), 3(brown)
The pink graph, where b=0, is a horizontal line at y=2 since anything to the 0th power is
equal to 1. This would give y=a(1)+c when b=0, and since a=1 and c=1, the graph would in
fact be a horizontal line at y=2.
There isn’t a common point for the different graphs but there is a point (0,1) that is central
to all of the graphs. I then graphed the function where b is a fraction. (The new blue
function below.)
When b is a fraction the graph only exists when x>0. As x approaches 0 from the right, the
function approaches the point (0,1) and as x approaches infinity the graph approaches
infinity. This point is the point central to all of the graphs.
Then I changed b to a negative (the new blue function below) fraction and saw that
the graph also only exists when x>0. This time, however, when x approaches 0 from the
right the graph goes to infinity and when x approaches infinity the graph approaches the
line y=1.
c.) Case 3: a, b are constant c varies: f(x) = a*xb+c | c = {-4, -3, -2, -1, 0, 1, 2, 3, etc}. Where a
and b are any constant number such that a and b are not equal.
First I looked at the affect “c” had on the graph when a=1 and b=1. I used the same color
coating as the previous case, except with values of c instead of b.
I can see that there is no common point between all of the graphs. The same graph is
shifted along the y-axis based on the value of c. Next I changed the value of b to see if that
would affect anything.
Again the graphs have no point in common. The graphs again are shifted up and down
along the y-axis with the different values. Next I set the value of a=2 to see what effect the
different values of c have on the function.
Reflection on Exploration:
Using Ti-Nspire was beneficial to fully see how the different variables affected a
power function. It was very helpful to be able to graph multiple functions on the same
coordinate plane. A feature of Ti-Nspire I also found to be useful was that you can hover
over one of the functions and a label will pop up reminding you which value of a, b, or c you
are looking at. Also, I wouldn’t normally be able to color coat the graphs to make the labels
of the graphs more distinct.