[table][tr][td][math]x^2+6x+8[/math][/td][td][math](x+2)(x+4)[/math][/td][td][math](x+3)^2-1[/math][/td][/tr][/table][size=150][br]Without graphing or doing any calculations, determine where the following features would be on a graph that represents the function.[br][/size][br]the vertex

the [math]x[/math]-intercepts

the [math]y[/math]-intercepts

[math](x+5)^2+1[/math]

[math](x-3)^2-7[/math]

[size=150]Think about the steps you took, and about reversing them. Try converting one or both of the expressions in standard form back into vertex form. Explain how you go about converting the expressions.[/size]

[size=150]Test your strategy by rewriting [math]x^2+10x+9[/math] in vertex form.[/size]

Does it appear that they define the same function?

If you convert your expression in vertex form back into standard form, do you get [math]x^2+10x+9[/math]?

[math]\displaystyle \begin {align} 3x^2 + 12x + 9 &\qquad& \text{Original expression}\\\\ 3(x^2 + 4x + 3) &\qquad& \phantom{222}\\\\ 3(x^2 + 4x + 3 + 1 - 1) &\qquad& \phantom{222}\\\\ 3(x^2+4x+4 -1) &\qquad& \phantom{222}\\\\ 3\left((x+2)^2 -1\right) &\qquad& \phantom{222}\\\\3(x+2)^2 - 3 &\qquad& \phantom{222} \end {align}[/math][br][br]What is the vertex of the graph that represents this expression?

Does the graph open upward or downward? Explain how you know.

[size=150]Rewrite each expression in vertex form. Show your reasoning.[/size][br][br][math]\text{-}2x^2-4x+6[/math]

[math]4x^2+24x+20[/math]

[math]\text{-}x^2+20x[/math]

Write [math]f(x)=2(x-3)(x-9)[/math] in vertex form without completing the square. (Hint: Think about finding the zeros of the function.) Explain your reasoning.

Write [math]g(x)=2(x-3)(x-9)+21[/math] in vertex form without completing the square. Explain your reasoning.

[table][tr][td]If your teacher gives you the data card:[/td][td]If your teacher gives you the problem card:[/td][/tr][tr][td][list=1][*]Silently read the information on your card.[/*][*]Ask your partner [br]“What specific information do you need?” [br]and wait for your partner to ask for information.[br]Only give information that is on your card.[br](Do not figure out anything for your partner!)[/*][*]Before telling your partner the information,ask [br]“Why do you need to know (that piece of information)?”[/*][*]Read the problem card, and solve the problem independently.[/*][*]Share the data card, and discuss your reasoning.[/*][/list][/td][td][list=1][*]Silently read your card and think about what [br]information you need to answer the question.[/*][*]Ask your partner for the specific information [br]that you need.[/*][*]Explain to your partner how you are using [br]the information to solve the problem.[/*][*]When you have enough information, share the [br]problem card with your partner, and solve the problem independently.[/*][*]Read the data card, and discuss your reasoning.[/*][/list][/td][/tr][/table]Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

[table][tr][td][math](x+5)(x+3)[/math][/td][td][math]x^2+8x+15[/math][/td][td][math](x+4)^2-1[/math][/td][/tr][/table][br]Select all of the statements that are true about the graph of this function.

[table][tr][td][math](x-4)(x+6)[/math][/td][td][math]x^2+2x-24[/math][/td][td][math](x+1)^2-25[/math][/td][/tr][/table][br]What is the [math]y[/math]-intercept of the graph of the function?[br]

What are the [math]x[/math]-intercepts of the graph?

What is the vertex of the graph?

[table][tr][td][math]x^2-7x+6[/math][/td][td][math]\text{original expression}[/math][/td][/tr][tr][td][math]x^2 - 7x + \left(\text-\frac{7}{2}\right)^2 + 6 -\left(\text{-} \frac{7}{2}\right)^2[/math][/td][td][math]\text{step 1}[/math][/td][/tr][tr][td][math]\left(x-\frac{7}{2}\right)^2+6-\frac{49}{4}[/math][/td][td][math]\text{step 2}[/math][/td][/tr][tr][td][math]\left(x-\frac{7}{2}\right)^2+\frac{24}{4}-\frac{49}{4}[/math][/td][td][math]\text{step 3}[/math][/td][/tr][tr][td][math]\left(x-\frac{7}{2}\right)^2-\frac{25}{4}[/math][/td][td][math]\text{step 4}[/math][/td][/tr][/table][br]In step 1, where did the number [math]\text{-}\frac{7}{2}[/math] come from?[br]

In step 1, why was [math]\left(\text-\frac72\right)^2[/math] added and then subtracted?[br]

What happened in step 2?

What happened in step 3?[br]

What does the last expression tell us about the graph of a function defined by this expression?

[math]d(x)=x^2+12x+36[/math]

[math]f(x)=x^2+10x+21[/math]

[math]g(x)=2x^2-20x+32[/math]

Give an example that shows that the sum of two irrational numbers can be rational.

Give an example that shows that the sum of two irrational numbers can be irrational.

Give an example that shows that the product of two irrational numbers can be rational.

Give an example that shows that the product of two irrational numbers can be irrational.

[size=150]Select [b]all[/b] the equations with irrational solutions.[/size]

[size=150]What are the coordinates of the vertex of the graph of the function defined by [math]f(x)=2(x+1)^2-4[/math]?[/size]

Find the coordinates of two other points on the graph.[br]

[size=150]How is the graph of the equation [math]y=(x-1)^2+4[/math] related to the graph of the equation [math]y=x^2[/math]?[/size]