11-02: Quadratic Functions

Ordered pair: a set of independent (x) values that represent a relationship

Relation: a set of inputs (x) and outputs (y)

Function: a relation with one output (y) for each input (x) and no x value may be repeated

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Identifying Functions

With points: no x values can be the same or repeat

With a graph: must pass the vertical line test

→ Vertical line test: draw a vertical line through the graph, cutting through it. If it intersects at more than 1 point then it fails the test and thus is not a function

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Function Notation

The way that a function is written → f(x) is essentially our new y

f(x) can also be a value, f(value), and you would then substitute that value in for x

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Domain and Range of Functions

• Domain: every x value your function is allowed to be
• Range: every y value your function is allowed to be

Use:

< less than

≤ less than or equal to

to set these boundaries of what your x and y values can be respectively

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Properties of Quadratic Functions

The quadratic function becomes a U-shaped parabola when graphed

Forms:

• Standard Form:

f(x) = ax² + bx + c where a ≠ 0

• c is y intercept
• Cannot graph from this form
  • Factored Form:

f(x) = a(x-r)(x-s)

• x intercepts are (r,0) and (s,0)
• To find the y intercept, let x = 0
• Can graph from this form
  • Vertex Form:

f(x) = a(x-h)² + k

• Vertex is (h,k)
• To find y intercept, let x = 0
• Can graph from this form

Direction of Opening, a’s Sign:

• If a is negative, parabola opens downwards and has a maximum (max frowns and is negative)
• If a is positive, parabola opens upwards and has a minimum

The min or max value is also the y value of the vertex

Axis of Symmetry (AOS):

The value of the x value at the vertex of the parabola (at which the function is at its min or max value)

Formula: (r + s)/2

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Profit and Revenue

Revenue: the total amount of money taken in

Cost: the amount of money paid for goods or expenses

Profit: the amount of money left after expenses → revenue - cost

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Revenue

Revenue = selling price x number of items

1. Let x represent the number of [@@value@@] increases or decreases
2. Chart:

If it is an increase, +

If it is a decrease, -

3. Find x intercepts → (new selling price)(new number of items) → solve by making it equal to zero
4. Find AOS by adding both points and dividing by 2 → (r+s)/2
5. Find new selling price or number of items by substituting in your new AOS value into either “new” equation on the chart (depending on what you want to find)

Profit

Profit = profit per item x number of items

1. Let x represent the number of [@@value@@] increases or decreases
2. Chart:

3. Find x intercepts → (Profit per item)(number of items) and solve by making x = 0

4. Find AOS by adding both points and dividing by 2 → (r+s)/2
5. Substitute back into chart “new” equations for new Profit Per Item or new Number of Items

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Solving Quadratic Equations

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• Factoring: Make the equation = to zero and factor, take your factored brackets and make them = to zero, isolate x

• Quadratic formula: make the equation = to zero and plug into the following formula

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  Zeroes of a Quadratic Function

  The radicand in quadratic formula is the discriminant → %%b² - 4ac%%

  The value of the discriminant is an indication of the number of solutions

• Positive Discriminant: 2 solutions, real and unequal roots

• Discriminant = Zero: 1 solution, real and equal roots

• Negative Discriminant: no solutions, imaginary roots

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  Discriminant with K

  1. Find K by making D=0 → becomes a new quadratic
  2. Draw a chart to find D>0 and D<0 (remember: the value of something that has been square root’ed is both positive and negative)

                               <                                  <                        <                                   <                       <

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Linear Quadratic Systems

Solve algebraically:

1. Make both expressions = to each other
2. Smoosh both equations together (make equal to zero) and solve y
3. Get your x value of the point
4. Find the y value of the point by substituting x into either equation

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