Download LectureSection2.2

2.2 Finding Quadratic Equations
In section 1.2 you learned how to generate linear equations when we had ordered
pairs. We used the format y = mx + b. We’re going to do something similar for
quadratics.
Recall from 2.1 we talked about quadratic equations in the form:
y  ax 2  bx  c
There is another format we can use that is
really helpful when we have the vertex. It is:
y  a( x  h)2  k
Where (h,k) is the vertex
To complete the equation, we put in the
values for “x” and “y” using a given ordered
pair, substitute the “h” and “k” values, then
solve for “a.” Once we get “a”, we can rewrite the equation using the values for
“a”, “h”, and “k”.
y  a ( x  h) 2  k
Your text refers to this format as STANDARD FORM.
I know we are going to be relying on the calculator to determine equations, but
it’s important that you know how to generate a quadratic by hand.
Review of Example 2.2.2 in your text.
Example 2.2.2: Finding a Quadratic Equation
Consider the data regarding the number of morning newspaper companies
in the United States since 1940. Find an equation by hand to model the
data. Then use the equation to predict the number of morning papers
there will be in 2025. For simplicity we can consider 1940 to be year 0.
Solution:
Having graphed the data, you can see the point (20,312) is a reasonable estimate for the vertex,
substituting into the equation y = a(x - h)2 + k we have: y = a(x - 20)2 + 312.
The graph passes through the point (65,817). Substituting we have:
817 = a(65 - 20)2 + 312
substitute 65 for x and 817 for y
817 = 2025a + 312
simplifying (65 - 20)2
505 = 2025a
subtract 312 from both sides
a ≈ .249
divide both sides by 2025
The equation y = .249(x - 20)2 + 312 is a fairly good model for the data. Use it to predict for year 2025.
y = .249(x - 20)2 + 312
year 2025 would mean x = 85
2
y = .249(85 - 20) + 312
replace x with 85
2
y = .249(65) + 312
parenthesis first
y = .249(4225) + 312
y = 1052.025 + 312
y = 1364.025
exponents next
multiplication next
add
Final Answer: If the pattern of growth continues in a similar fashion, there will be approximately 1364
morning newspaper companies in 2025.
Finding an equation by hand using the standard form of a parabola is a confidence builder, but as we
saw in the last chapter using regression is much easier and more accurate.
Example 2.2.3 in your test takes the same data and uses the regression feature in
your calculator to get the equation.
If you read your text, you are supposed to ask me a question after example 2.2.3.
??????????
Results from 2.2.2
y  0.249( x  20)2  312
Results from 2.2.3
y  0.23x 2  8.13x  374.75
Practice 1
1. The chart shows the inches of drop in a bullet for different length shots for a
particular type of Hornady bullet design.
a) Use regression to find a quadratic equation to model the data, considering
drop as a function of yardage. Round the numbers in your equation to 4
decimal places.
b) Use your equation to predict the drop for a 215 yard shot, accurate to 1
decimal place.
c) Use your equation to predict the drop for a 500 yard shot, accurate to 1
decimal place.
Solutions:
a)
Enter into the calculator the data from the first two columns in the Hornady table, then select
the quad option to generate the regression equation: y  0.0013x 2  0.3166 x  20.3480
b)
Using the equation from a),
c)
Using the equation from a),
substitute 215 for x, solve for y.
substitute 500 for x, solve for y.
y  0.0013  (215) 2  0.3166  215  20.3480
y  0.0013  (500)2  0.3166  500  20.3480
y  12.3715
y  187.048
y  12.4 in
y  187.0 in
Practice 2
2. The golden gate is a suspension bridge where the road is
supported by equally spaced cables hanging from a large
parabolic cable, which is itself supported by 2 large towers.
a) Use the dimensions below (measured in feet) to find 3
ordered pairs that define the parabola, then regression to
find the equation for the parabola, accurate to 6 decimal places.
b) Use the equation to find the lengths of the 7 hanging cables between the left tower and the
40 foot center cable. Round to 2 decimal places.
Solutions:
a)
Establish 3 ordered pairs. Enter into the calculator the data from the
table, then select the quad option to generate the regression equation:
y  0.000132 x 2  0.633333x  800
b
The cables are 300 ft apart. Using the equation from a), substitute each
increment in for “x” and solve for “y” (height of each cable).
Distance From Left Tower (x) Length of Cable (y)
300 ft
600 ft
900 ft
1200 ft
1500 ft
1800 ft
2100 ft
Distance
Length of
From Left
Cable
Tower
0
2400
4800
800
40
800
621.88 ft
467.52 ft
336.92 ft
230.08 ft
147.00 ft
87.68 ft
52.12 ft
Homework: problems 2,6,8,9,10. Number 8 might cause you….to do some
thinking. Make sure you understand the question and that your answer(s) make
sense.