Rewrite each quadratic expression in standard form.

1. \((x+1)(7x+2)\)
2. \((8x+1)(x-5)\)
3. \((2x+1)(2x-1)\)
4. \((4+x)(3x-2)\)

Find the missing expression in parentheses so that each pair of quadratic expressions is equivalent. Show that your expression meets this requirement.

1. \((4x-1)(\underline{\hspace{1in}})\) and \(16x^2 -8x +1\)
2. \((9x + 2)(\underline{\hspace{1in}})\) and \(9x^2 -16x -4\)
3. \((\underline{\hspace{1in}})(\text-x + 5)\) and \(\text-7x^2 +36x-5\)

The number of downloads of a song during a week is a function, \(f\), of the number of weeks, \(w\), since the song was released. The equation \(f(w) = 100,\!000 \boldcdot \left(\frac{9}{10}\right)^w\) defines this function.

1. What does the number 100,000 tell you about the downloads? What about the \(\frac{9}{10}\)?

2. Is \(f(\text-1)\) meaningful in this situation? Explain your reasoning.

Consider the equation \(4x^2 - 4x -15 = 0\).

1. Identify the values of \(a\), \(b\), and \(c\) that you would substitute into the quadratic formula to solve the equation.

2. Evaluate each expression using the values of \(a\), \(b\), and \(c\).

   ​​​​\(\text- b\)

   \(b^2\)

   \(4ac\)

   ​​​​​\(b^2 - 4ac\)

   \(\sqrt{b^2 - 4ac}\)

   \(\text- b \pm \sqrt{b^2 - 4ac}\)

   \(2a\)

   \(\dfrac{\text- b \pm \sqrt{b^2 - 4ac}}{2a}\)

3. The solutions to the equation are \(x=\text-\frac 32\) and \(x=\frac52\). Do these match the values of the last expression you evaluated in the previous question?

1. Describe the graph of \(y=\text-x^2\). (Does it open upward or downward? Where is its \(y\)-intercept? What about its \(x\)-intercepts?)

2. Without graphing, describe how adding \(16x\) to \(\text-x^2\) would change each feature of the graph of \(y = \text-x^2\). (If you get stuck, consider writing the expression in factored form.)

   1. the \(x\)-intercepts
   2. the vertex
   3. the \(y\)-intercept
   4. the direction of opening of the U-shape graph