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Graphs of Quadratic Functions

Intercepts, vertex, axis of symmetry, and forms of a quadratic.

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Solving and Graphing Quadratic Functions

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Vocabulary

Fill in the empty boxes.

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

excellent for finding -intercepts, is: \begin{align*}f(x)=a(x-r_{1})(x-r_{2})\end{align*}

Factoring Polynomials

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

\begin{align*}ac = -30\end{align*}

\begin{align*}& 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)\end{align*}

________________________ \begin{align*}81x^4-16\end{align*} 

Difference of squares: \begin{align*}& 81x^4-16\\ & (9x^2-4)(9x^2+4)\end{align*}

Factor \begin{align*}9x^2-4\end{align*} with difference of squares also:

_______________________________

 

________________________  \begin{align*}6x^5-51x^3-27x = 0\end{align*}

Pull out Greatest Common Factor:

________________________________

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

_________________________________

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Remember:
 Pull out the Greatest Common Factor if there is one!

Graphing Quadratic Functions

Complete the following table.

Form Equation Used For
___________ \begin{align*}f(x)=ax^{2}+bx+c\end{align*} ________________________
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_____________________


Practice

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

  2. Consider the quadratic functions\begin{align*}y = 4x^2\end{align*}    \begin{align*}y = 5x^2\end{align*}    \begin{align*}y = 7x^2\end{align*} Which quadratic function would you expect to have the widest parabola? Explain your answer.

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

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