# 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 x -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 x -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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