Quadratic Functions

1
Quadratic Functions Systems of Equations

• Sections 3.3 3.4

2
Quadratic Functions

• A quadratic function in x is one that can be
  written in the form
• f(x) ax2 bx c with a ? 0.
• The graph of a quadratic function is called a
  parabola and may open up or down.
• The graph of a quadratic function is symmetric
  about a vertical line called the axis of
  symmetry.
• The vertex of the parabola is the point

a gt 0
a lt 0
3
Graphing Quadratic Functions

• Sketch the graph of f(x) 2x2 4x - 3
• First locate the vertex
• Find the y-intercept f(0) -3
• Graph the reflection of the y-intercept
• Find the x-intercepts Solve 2x2 4x - 3 0
• Locate other points if needed and sketch the
  graph.

-1
-5
4
Application of Quadratic Functions

• Ex. Find the maximum revenue.
• The demand function for the Brutus lawn chair
  company is given by p 32 - 0.2q. Find the
  quantity q which gives the maximum revenue.
• Revenue is given by r pq, so we get
• r (32 - 0.2q)q 32q - 0.2q2
• This is a quadratic function with a -0.2 and b
  32.
• So the maximum revenue occurs when
• The maximum revenue is then 32(80) - 0.2(80)2
  1280.

5
Systems of Equations

• A system of equations is a collection of two or
  more equations in two or more variables.
• Ex.
• A solution to a system of equations is a set of
  values that makes all the equations true at the
  same time.
• Graphically we can view a solution as any
  intersection of the graphs of the equations.
• We can check that the points (-3, -4) and (4, -3)
  are solutions to the system.

(4, -3)
(-3, -4)
6
Systems of Linear EquationsTwo-Variable Systems

• For a system of two linear equations, the graph
  of the system will consist of two lines.
• There are then three possibilities for the
  solution of such a system
• The two lines intersect in exactly one point.
• Then the system has one solution.
• The two lines are parallel and never intersect.
• The system has no solution.
• Both equations give the same line.
• The system has infinitely many solutions.

7
Solving Systems of Linear EquationsTwo Methods

• One method of solving a system of linear
  equations is called the elimination method.
• We eliminate one variable by adding a multiple of
  one equation to a multiple of the other.
• The other method is called the substitution
  method.
• We solve one equation for one variable and
  substitute this expression back into the other
  equation.

8
The Elimination Method

• Solve
• Begin by multiplying the first equation by 2
• Then multiply the second equation by -3
• Now add the two equations to get 17y -119
• We solve this to get y -7.
• Plug -7 in to get 3x 4(-7) -22
• So 3x - 28 -22 or x 2 and the solution is (2,
  -7).

9
The Substitution Method

• Solve
• Solve one equation for one variable in terms of
  the other
• Substitute this expression into the other
  equation
• Plug this value into either equation
• 5x - 2(4) -23, so 5x -15 or x -3.
• The solution is (-3, 4).

10
Brutus Lawn Chair Co.

• The Brutus lawn chair company makes two types of
  chairs, the standard deck chair requiring 15
  yards of webbing and 20 ft of tubing, and the
  deluxe lounge chair requiring 24 yards of webbing
  and 18 ft of tubing. If they have 900 yards of
  webbing and 990 feet of tubing available, how
  many can they make of each.
• Let d be the number of deck chairs and l be the
  number of lounge chairs.
• We need 15d 24l yards of webbing and 20d 18l
  ft of tubing.
• So the system we want to solve is

11
Brutus Lawn Chair Co., cont.

• Solve
• Multiply the first equation by 4 and the second
  by -3
• Add the two equations to get 42l 630
• So l 15.
• Then 15d 24(15) 900.
• 15d 540
• d 36
• They can make 36 deck chairs and 15 lounge
  chairs.

12
Key Suggested Problems

• Sec. 3.3 15, 17, 19, 21, 25, 27, 29
• Sec. 3.4 1, 3, 9, 11, 13, 15, 17, 25, 33, 37