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Quadratic Functions
Graphing Quadratics in Vertex Form
Using the graph of f(x) = x2 below and what you know of transformations, describe the graph of
f(x) = (x + 4)2 – 2 and then sketch the graph.
A ______function is a polynomial function with degree 2.
The parent function is ______.
The graph of a quadratic function is a ______.
The vertex form of a quadratic function:
fx= a(x-h)2+ k where a ≠ 0 and (h, k) is the ______.
When a > 0, the parabola opens ______and the function has a ______.
When a < 0, the parabola opens ______and the function has a ______.
Axis of symmetry: line where ______.
Steps to graphing a quadratic function in vertex form:
1. Identify the vertex and plot.
2. Plot another point using x = h + 1
3. Plot a symmetric point
4. Sketch the curve
Graph the functions, state the domain and range:
Writing a Quadratic Function in Vertex Form
1. Identify vertex: (h, k) =
2. Choose another point on the graph (x, y) =
3. Substitute h, k, x, and y into
fx= a(x-h)2+ k and solve for a
3. 4.
Graphing Quadratics in Standard Form
fx=ax2+ bx+c is the ______form of a ______.
The ______coordinate of the vertex is - b2a
The axis of symmetry is the line______.
Steps to graphing a quadratic function in standard form:
1. Identify the vertex and plot.
2. Plot another point using and x close to the vertex
3. Plot a symmetric point
4. Sketch the curve
Identify the vertex, the axis of symmetry, the maximum or minimum value, and the domain and range:
1. 2.
Quadratic Inequalities
· Graph the function (< or > is dashed/ ≤ or ≥ solid)
· Shade above the curve for greater than and below the curve for less than
4)
Solving problems using graphs of quadratic functions
Quadratic functions have either a ______or a ______.
The maximum or minimum value is always at the ______.
To solve these problems, draw a sketch of the graph (or graph on your calculator) and think!
Picture what is happening in the problem.
· If the problem asks for a maximum or minimum, look at the vertex
· If the problem asks for a value at a certain time—plug that time into the function (time is usually x)
· If the problem asks when an object will hit the ground (or when the function will be zero) look at the x intercept(s).
Example 1:
The height in feet of a rocket above the ground is given by h = – 16t2 + 64t + 190 for t ≥ 0 where t is the time in seconds. Find the elevation of the launch, the time it takes the rocket to reach maximum height, the maximum height of the rocket, and the time the rocket is in the air.
a. Elevation of the launch:
What is the time at launch?
b. Time to reach maximum height:
Which variable is time? At what point is height at a maximum?
c. Maximum height:
d. Time rocket is in the air:
Where on the graph is the rocket in the air?
Example 2:
You have designed a new style of sports bicycle! Now you want to make lots of them and sell them for profit. The amount of profit you earn will depend on the price you charge. If you set the price too high, you will not sell many bikes, but if you charge too little, you may sell a lot of bikes but won’t make much money.
Based on your research you estimate your profit will be P = -200p2 + 92,000p - 8,400,000
What should you charge for each bike? What do the x-intercepts represent on the graph of this function?
Example 3:
The number of bacteria in a refrigerated food is given by N(T) = 20 T2 – 20T + 120, for – 2 ≤ T≤ 14 and where T is the temperature of the food in Celsius. What is the best temperature to keep bacteria to a minimum?
Example 4:
The cross-section of a large radio telescope used to contact space aliens is a parabola. The dish is set into the ground. The equation that describes the cross-section is
d= 275x2- 43x- 323 where d gives the depth of the dish below ground and x is the distance from the control center, both in meters. If the dish does not extend above the ground level, what is the diameter of the dish?
Example 5:
You and a friend are hiking in the mountains. You want to climb to a ledge that is 20 ft. above you. The height of the grappling hook you throw is given by the function h(t) = -16t2 – 32t + 5.
a) What is the maximum height of the grappling hook?
b) Can you throw it high enough to reach the ledge?
Quadratic Systems
Linear-Quadratic Systems
These systems have 1 ______function and 1 ______function.
The points where graphs of the functions intersect are the ______of the system.
To solve these systems by graphing:
· Graph the parabola
· Graph the line
· The intersection point(s) are the solution(s)—if the curves don’t intersect, write Ø
· You can check your solution(s) by substituting them into both equations to see if they work.
Example:
Find the solution to the system of equations: y = -2x + 3 y = x2 – 4x – 5
Use the 2nd Calc -> intersect function
Practice: Find any solutions to the following systems
1. y = x2 + 6x + 3 y = x – 7 2. y = x2 – 4 y = –x – 2
Quadratic-Quadratic Systems
You can use the same method to solve systems of two quadratics.
3. y = x2 – 4 y = –x2 – 2x 4. y = 2x2 – 3x – 2 y = –x2 – 2
Real world problem: You have decided to open up a coffee shop in your neighborhood. As the owner of the coffee shop, you need to determine how many workers you will need at any time. Each employee can only work one shift a day, and is paid $40 for that shift (this does not include tips).
Write a linear equation to model the relationship between the number of employees working (x) and the amount of money (y) you will be paying in wages that day.
You have hired a team of analysts to come into study how much money your coffee shop is making based on the number of employees working for that day. After studying your shop, they have found that the following quadratic equation to model the amount of money your coffee shop makes (y) based on the number of employees working that day (x):
Earnings: y = -10x2 + 100x
Look at the graph of this situation:
1. For what number of employees will your expenses be equal to your income?
2. If you schedule more workers, will you make money or lose money?
3. Looking at the graph of this situation, what number of workers will maximize your profits?