Clare says, “I know that \(\sqrt3\) is an irrational number because its decimal never terminates or forms a repeating pattern. I also know that \(\frac29\) is a rational number because its decimal forms a repeating pattern. But I don’t know how to add or multiply these decimals, so I am not sure if \(\sqrt3 + \frac29\) and \(\sqrt3 \boldcdot \frac29\) are rational or irrational."

1. Here is an argument that explains why \(\sqrt3 + \frac29\) is irrational. Complete the missing parts of the argument.

   1. Let \(x = \sqrt3 + \frac29\). If \(x\) were rational, then \(x - \frac29\) would also be rational because . . . .
   2. But \(x - \frac29\) is not rational because . . . .
   3. Since \(x\) is not rational, it must be . . . .
2. Use the same type of argument to explain why \(\sqrt3 \boldcdot \frac29\) is irrational.

The following expressions all define the same quadratic function.

1. What is the \(y\)-intercept of the graph of the function?
2. What are the \(x\)-intercepts of the graph?
3. What is the vertex of the graph?
4. Sketch a graph of the quadratic function without using technology. Make sure the \(x\)-intercepts, \(y\)-intercept, and vertex are plotted accurately.

Here are two quadratic functions: \(f(x) = (x + 5)^2 + \frac12\) and \(g(x) = (x + 5)^2 + 1\).

Andre says that both \(f\) and \(g\) have a minimum value, and that the minimum value of \(f\) is less than that of \(g\). Do you agree? Explain your reasoning.

Without using graphing technology, sketch a graph that represents each quadratic function. Make sure the vertices and any intercepts that fit on the given grids are plotted accurately.

\(f(x) = x^2 + 4x + 3\)

\(g(x)=x^2-4x+3\)

\(h(x) = x^2 - 11x + 28\)