2.1 The Shape of a Quadratic Equation
In chapter 1 the focus was on lines. All of the data we collected was pretty linear whereby we could draw in a trend line. Then we used the calculator’s regression feature to get a Linear Equation.
Well the world is not all “linear”. Chapter 2 is all about “parabolic” shapes (all about “parabolas”).
The definition of a parabola is a symmetrical plane curve that forms when a cone intersects with a plane parallel to its side.
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Remember, we can “model” any geometric shape.
Recall the format for linear equations was y = mx + b?
Well, the format for an equation that models a parabola is called a “quadratic” equation and the format looks like this: y = ax2 + bx + c
This you must know.
As your book discusses, there are tons of uses that apply parabolas and thus quadratic equations. This means that, given the empirical data, the graph will take the shape of a parabola.
Practice 1
- American newspaper circulation enjoyed continuous growth until the 1970’s where it roughly leveled off then began a steady decline in the 1990’s, presumably due to the internet.
a)Make a graph of the data (use graph paper, label completely, and choose the correct axis for the independent(x) and dependent(y) variables).
b)Find the slope between 1940 and 1945. Explain the meaning of the slope in context.
c)Find the slope between 1960 and 1965. Explain the meaning of the slope in context.
d)Find the slope between 1995 and 2000. Explain the meaning of the slope in context.
e)Add a trend line to the graph.
f)Estimate the vertex and explain its meaning in context.
g)Estimate both x-intercepts and explain their meaning in context.
Solutions:
a,e)
b)c)d)
f)Answers will vary.
The vertex is estimated to be (1976,63000) meaning 63,000 is the maximum circulation and 1976 the year it occurred.
g)Answers will vary.
The x-intercepts are estimated to be:
(1912, 0) meaning 1912 is the approximate year that newspapers began to circulate.
(2042, 0) meaning 2042 is the approximate year that newspapers will no longer circulate.
Practice2
5. A satellite dish is a precise mathematical shape (a parabola) that has the property of focusing a reflected signal at a single point. The scale drawing of a cross section of a 28 foot wide parabolic satellite dish is shown on the graph where each square is 1ft x 1ft. The dish is designed using the equation y = .04x2 + 2. The ordered pair solutions of the equation become the dimensions necessary to build this amazing device.
a)Treat the x-axis as the ground and use the equation to find the height of the dish above the ground at the 2 foot intervals shown on the graph.
b)Recall that the quadratic equation
y = ax2 + bx + c defines the geometric parabola. It turns out that the focus point is located a distance above the bottom of the dish. Find this distance for this satellite dish.
Solutions:
Substitute each x value (from -14 to 14) into the equation and solve for y (the height above the x-axis).Horizontal Distance From Vertex / Height
Above x-axis
-14 / 14 / 9.84 ft
-12 / 12 / 7.76 ft
-10 / 10 / 6.00 ft
-8 / 8 / 4.56 ft
-6 / 6 / 3.44 ft
-4 / 4 / 2.64 ft
-2 / 2 / 2.16 ft
0 / 2.00 ft
a)b)
Homework: 2,3, 6
When you do your homework for problem #6, use the graph paper on the next page. This way everyone will have the same size and I/you can check it easily. Make sure you follow the directions.