What is an equation for the line
that passes through the point \((2,3)\)
and is parallel to the line
given by the equation \(y = -\frac{1}{2}x+7?\)
Protip: it might help the sketch
the initial line, the point,
then the new line.
What is an equation for the line
that passes through the point \((-3,5)\)
and is perpendicular to the line
given by the equation \(y = -\frac{1}{2}x+7?\)
Protip: it might help the sketch
the initial line, the point,
then the new line.
Without appealing to technology,
explain, as if convincing a skeptical peer in the class,
how you can tell that the three points
\((-1,3)\) and \((1,-1)\) and \((5,-9)\)
all lie on the same line.
Suppose a line intersects the \(x\)-axis at the number \(12,\)
and the area of the triangle formed by this line
and the positive \(x\)-axis and \(y\)-axis is \(30.\)
What must the slope of this line be?
The price of Johnson & Johnson
(NYSE:JNJ) stock
was $51.66 per share on 1 January 2004.
Look up the price that JNJ closed at on the first day of this month.
What was the average monthly change in share price
between these two dates?
What was the average annual change in share price
between these two dates?
According to the US Energy Information Administration (EIA),
the average price of mid-grade conventional retail gasoline in Colorado
was $2.59 at the beginning of January 2021.
Look up the average price of gas in Colorado for the first of this month.
What was the average monthly change
in gas price between these two dates?
Using this last figure, write down an equation \(p(t)\)
of a linear function \(p\) that models the price of gas
as a function of \(t\) months since January 2021.
According to this model \(p\) what was the price of gas
at the beginning of July 2023?
According to this model \(p\) what was the price of gas
at the beginning of January 2000?
According to this model \(p\)
when will the price of gas be over $6 per gallon?
We built this model using only two data points,
so maybe its not a very accurate model.
Plot the graph \(y = p(t)\) over a scatterplot
of the following data from EIA.
2021
Month
Price
January
$2.59
April
$3.25
July
$3.80
October
$3.88
2022
Month
Price
January
$3.60
April
$4.24
July
$5.10
October
$3.90
2023
Month
Price
January
$3.71
April
$3.80
July
$4.21
October
$4.00
2024
Month
Price
January
$3.01
April
$3.39
July
$3.65
October
$3.62
Visually, does \(p\) appears to be an accurate model?
If not, try to adjust the parameters of the model,
the slope and \(y\)-intercept,
to make it fit the data better.
Do you think it’s a reasonable idea to use \(p\)
as a model for gas prices in the future?
I.e. do you think \(p\) accurately predicts
gas prices in the year 2050? or 2100?