A-9.4 (TE): Applying Graphs of Quadratic Functions
occupation overview
key terminology
1. Characteristics of a Parabola
2. QUADRATIC FUNCTIONS IN INTERCEPT FORM
3. QUADRATIC FUNCTIONS IN VEREX FORM
CAREER SPOTLIGHT
Section 1 of 13
Aerospace Engineers
Aerospace engineers design primarily aircraft spacecraft, satellites, and missiles.
Career Spotlight
- Discuss aerospace engineering with students by
reading the Career Spotlight together. - Find local colleges and universities with an aerospace
engineering program to share with students. - Research local companies that employ aerospace
engineers, and ask what they do for the companies.
As Essential Workers
- Have students consider why this career would be considered essential in a time of emergency.
- Location and other mitigating factors impact who is considered an “essential worker.” Discuss with your students what other careers might be considered essential in your area.
Learn More!
- They create and test prototypes.
- They may develop new technologies for use in aviation, defense systems, and spacecraft.
- Aerospace engineers often become experts in one or more related fields: aerodynamics, thermodynamics, materials, celestial mechanics, flight mechanics, propulsion, acoustics, and guidance and control systems.
- They typically specialize in one of two types of engineering: aeronautical or astronautical.
Career Cluster
Career Pathway
Education Needed
Entry-level aerospace engineers usually need a bachelor’s degree. Bachelor’s degree programs include classroom, laboratory, and field studies in subjects such as general engineering principles, propulsion, stability and control, structures, mechanics, and aerodynamics, which is the study of how air interacts with moving objects.
Section 3 of 13
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Career Story
Watch a video:
- Have students consider why this career would be considered essential in a time of emergency.
- Location and other mitigating factors impact who is considered an “essential worker.” Discuss with your students what other careers might be considered essential in your area.
Make a Connection
Section 4 of 13
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Connecting Aerospace Engineers and Algebra
Lesson Objective
In this lesson, you will look at how an aerospace engineer uses quadratic functions to model the path of a flight or the forces acting on an aircraft.
Algebra Concepts
- Graph quadratic functions.
- Use factoring and completing the square to identify characteristics of parabolas.
- Draw students’ attention to any names or images related to the careers. Does the person represented here fit their mental model of who should be performing this career? Why or why not?
- Use the equity data on the Labor Market Navigator* to look up more detailed information about the career presented. What does the data say? Are some groups of people represented more or less than others? Why do you think that is?
*If your state has not purchased the P2C Labor Market Navigator, you may access equity data through US Bureau of Labor Statistics (opens in a new tab).
Section 5 of 13
Graphing a Quadratic Function
- The graph of a quadratic function is a parabola. It has a line of symmetry that passes through the vertex. A parabola can have 0, 1, or 2 x-intercepts.
- For a parabola that opens down, the y-coordinate of the vertex is the maximum value of the function.
- For a parabola that opens up, the y-coordinate of the vertex is the minimum value of the function
- For the function shown in the graph, the vertex is (5, −4), and −4 is the minimum.
- The vertex form of a quadratic function with vertex (h, k) is f(x) = a(x – h)2 + k.
- The vertex form of the function shown is f(x) = (x – 5)2 – 4.
- The intercept form of a quadratic function with x-intercepts p and q is
f(x) = a(x – p)(x – q). - The intercept form of the function shown isf(x) = (x – 3)(x – 7).The x-intercepts are 3 and 7.
Section 6 of 13
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Characteristics of a Parabola
Imagine this.
When a spacecraft travels beyond Earth’s atmosphere, the gravitational force is close to zero. To simulate microgravity on Earth, aerospace engineers use a special type of aircraft. The pilot flies the aircraft upward and then cuts the thrust, allowing the aircraft to continue rising and then fall back down in a parabolic pattern before increasing the thrust again.
The graph shows the height in meters of the aircraft as a function of time in seconds after the pilot begins to fly upward. The height is measured relative from when the pilot cuts the thrust. The solid line indicates the part of the flight that is parabolic, when the gravity is near zero. Identify the vertex and x-intercepts of the parabola, and interpret what each represents in this situation. Then write a quadratic function in intercept form for the parabola.
Devise a plan.
- Identify and interpret the vertex.
- Identify and interpret the x-intercepts of the parabola.
- Write the function for the parabola.
Walk through the solution.
Identify and interpret the vertex.
Identify and interpret the x-intercepts of the parabola. The vertex is (30, 800). It represents the maximum height of the jet, 800 meters above the point where the pilot cut the thrust. The jet reaches that point 30 seconds after the beginning of the maneuver.
- What does y represent in this situation?
- What does x represent?
Identify and interpret the x-intercepts of the parabola.
The x-intercepts are 20 and 40. They represent the beginning and ending times of the parabolic portion of the flight. The pilot cuts the thrust 20 seconds into the maneuver and increases the thrust again at 40 seconds.
- In Step 2, how can you use the intercepts to determine the duration of the maneuver?
Write the function for the parabola.
To write a function for the parabola in intercept form, substitute 20 and 40 for p and q in
f(x) = (x − p)(x − q).
f(x)
= a(x − p)(x − q)
= a(x − 20)(x − 40)
Then use the vertex, (30, 800), to solve for a.
800 = a(30 − 20)(30 − 40)
800 = a(10)(−10)
800 = −100a
−8 = a
The function in intercept form for the parabola is
f(x) = −8(x − 20)(x − 40).
- Before finding the value of a in Step 3, how can you tell from the graph whether the value will be positive or negative?
- Ask students to sketch a parabola that models the same flight but with x= 0 representing the moment that the pilot cuts the thrust. In other words, the parabola is shifted to the left by 20 seconds.
- What are the x-intercepts and vertex of thenew parabola?
- What is the function in intercept form for thenew parabola?
Answers: x-intercepts: 0, 20; vertex: (10, 800); f(x)=−8x(x− 20)
Section 7 of 13
Section 8 of 13
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Quadratic Functions in Intercept Form
Imagine this.
An aerospace engineer represents the height in meters of an aircraft during a parabolic maneuver, relative to the height when the pilot reduces the thrust, using the function f of x equals negative 8 times open paren x minus 20 close paren times open paren x minus 40 close paren, where x is the time in seconds. Suppose the pilot cuts the thrust when the aircraft is at an altitude of 9000 meters. The engineer writes a new function where y represents the altitude of the aircraft. What is the function in
intercept form for the altitude of the jet that the engineer writes? Then graph the new function.
Devise a plan.
- Write a function for the altitude of the aircraft.
- Rewrite the function in intercept form.
- Graph the function.
Write a function for the altitude of the aircraft.
For any height given by the original function, the altitude of the aircraft is 9000 meters greater. So, the altitude is given by the function f(x) = -8(x – 20)(x – 40) + 9000.
- In Step 1, why is the altitude 9000 meters greater than the height given by the original function?
To rewrite the function in intercept form, first simplify and then factor.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
f(x) = -8(x – 20)(x – 40) + 9000.
Graph the function.
To graph the function, first plot the intercepts at (−5, 0) and (65, 0). Then locate the line of symmetry halfway between the intercepts, x = 30. Substitute 30 for x into the function to find the y-coordinate of the vertex.
f(x) = −8(x + 5)(x − 65)
= −8(30 + 5)(30 − 65)
= −8(35)(−35)
= 9800
Plot the vertex at (30, 9800). Use the vertex,
intercepts, and line of symmetry to draw the
parabola.
- In Step 3, what does the vertex of the parabola represent?
- Do the x-intercepts represent times that the aircraft is on the ground?
- What part of the graph actually represents the altitude of the jet?
Section 9 of 13
Apply Quadratic Functions in Intercept Form
Complete the following:
Use this formula sheet if needed.
Section 10 of 13
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Quadratic Functions in Vertex Form
Imagine this.
In order to design aircraft, aerospace engineers need to
understand the forces that act on an object as it moves through the air. Thrust is the force that moves an airplane forward. Drag is
the force that works against thrust, slowing the airplane down.
An aerospace engineer writes the function
f(x) = 0.15×2 – 27x + 1500
for the drag y, in pounds, of an aircraft as a function of airspeed x,
in knots. At which airspeed is there the least amount of drag?
Devise a plan.
- Rewrite the function in vertex form.
- Graph the function.
- Find the airspeed that produces the minimum amount of drag.
Graph the function.
To graph the function, first plot the vertex at (90, 285). Then substitute a value for x to find another point on the graph.
f(x) = 0.15×2 − 27x + 1500
= 0.15(x2 − 180x) + 1500
= 0.15(x2 − 180x + 8100) + 1500 − 0.15(8100)
= 0.15(x − 90)2 + 285
Rewrite the function in vertex form by completing the square.
f(x) = 0.15×2 − 27x + 1500
= 0.15(x2 − 180x) + 1500
= 0.15(x2 − 180x + 8100) + 1500 − 0.15(8100)
= 0.15(x − 90)2 + 285
= 660
The graph passes through the point (40, 660). Draw the left half of the parabola through (40, 660) and ending at the vertex. The line x = 90 is a line of symmetry, so draw the right half of the parabola to mirror the left.
Step 2 shows that the point (40, 660) lies on the graph. What other point on the graph has a y-coordinate of 660? How do you know?
Step 2 shows that the point (40, 660) lies on the graph. What other point on the graph has a y-coordinate of 660? How do you know?
Find the airspeed that produces the minimum amount of drag.
The vertex represents the minimum value of the function. The least amount of drag on the plane is 285 pounds, when the airspeed is 90 knots.
- Ask students to substitute the expression
x+ 10 for x in the vertex form of the function
and then simplify. Have them graph their new
function in the same coordinate plane with the
original graph. - What is the vertex of the new parabola? How
do the graphs compare to one another? To
graph the original function, the point (40, 660)
was plotted. For what value of x does the new
function have the value y= 660?
Answers:
(80, 285); The new graph is the original shifted left 10 units. When x= 30, y= 660
Section 11 of 13
Section 12 of 13
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Career Spotlight
Section 13 of 13
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Career Spotlight: Check
Complete and submit this assignment to receive a score and feedback from your teacher.
Use this formula sheet if needed.