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Graphs of Quadratic Functions

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Solving and Graphing Quadratic Functions

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Word Definition
___________ The characteristic shape of a quadratic function graph, resembling a bowl or the silhouette of a bell
Vertex ______________________________________________________________
Quadratic Form ______________________________________________________________
___________ f(x)=ax^{2}+bx+c
Standard Form ______________________________________________________________

excellent for finding -intercepts, is: f(x)=a(x-r_{1})(x-r_{2})

Factoring Polynomials

Choose a name for each method to help you remember. Also, fill in any blanks.
Method Equation Example Steps
________________________ 2x^4-x^2-15

ac = -30

& 2x^4-x^2-15\\& 2x^4-6x^2+5x^2-15\\& 2x^2(x^2-3)+5(x^2-3)\\& (x^2-3)(2x^2+5)

________________________ 81x^4-16 

Difference of squares: & 81x^4-16\\& (9x^2-4)(9x^2+4)

Factor 9x^2-4 with difference of squares also:



________________________  6x^5-51x^3-27x = 0

Pull out Greatest Common Factor:


Use a combination of the first two methods to factor the rest:


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 Pull out the Greatest Common Factor if there is one!

Graphing Quadratic Functions

Complete the following table.

Form Equation Used For
___________ f(x)=ax^{2}+bx+c ________________________
Vertex Form ________________________ ________________________
___________ ________________________

finding -intercepts

Answer the following questions:
  1. Which letter of which form is the y-intercept? _____________________
  2. Where is the vertex in vertex form_____________________
  3. How do you find the vertex in standard form_____________________
  4. What is the axis of symmetry and how do you find it from standard form_____________________


Factor the following completely:
  1. x^4-4x^2-45
  2. 4x^4-11x^2-3
  3. 16x^4-1
  4. 6x^5+26x^3-20x
  5. 625-81x^4
Answer the following questions:
  1. Which direction does a parabola open if the leading coefficient ( ) is negative?

  2. Consider the quadratic functionsy = 4x^2    y = 5x^2    y = 7x^2 Which quadratic function would you expect to have the widest parabola? Explain your answer.

Sketch the graph of each function:
  1. y = -x^2 + 7
  2. y = 3x^2 + 6x +1
  3. y = \frac{1}{2} x^2 + 2x +4
  4. y = (x + 6)^2 - 3
  5. y = -x^2 -8x -17

Click here for answers.

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