The graph that represents \(f(x) = (x+1)^2-4\) has its vertex at \((\text-1,\text-4)\).

Explain how we can tell from the expression \((x+1)^2-4\) that -4 is the minimum value rather than the maximum value of \(f\).

Each expression here defines a quadratic function. Find the vertex of the graph of the function. Then, state whether the vertex corresponds to the maximum or the minimum value of the function.

1. \((x - 5)^2 + 6\)
2. \((x + 5)^2 - 1\)
3. \(\text- 2(x+3)^2 - 10\)
4. \(3(x - 7)^2 + 11\)
5. \(\text- (x - 2)^2 - 2\)
6. \((x + 1)^2\)

Consider the equation \(x^2=12x\).

1. Can we use the quadratic formula to solve this equation? Explain or show how you know.
2. Is it easier to solve this equation by completing the square or by rewriting it in factored form and using the zero product property? Explain or show your reasoning.

Match each equation to the number of solutions it has.

Which equation has irrational solutions?

Let \(I\) represent an irrational number and let \(R\) represent a nonzero rational number. Decide if each statement is true or false. Explain your thinking.

1. \(R \boldcdot I\) can be rational.
2. \(I \boldcdot I\) can be rational.
3. \(R \boldcdot R\) can be rational.

Here are graphs of the equations \(y=x^2\), \(y=(x-3)^2\), and \(y=(x-3)^2 + 7\).

1. How do the three graphs compare?

   

   A parabola labeled y equals x squared in x y plane, origin O. X axis  negative 6 to 8, by 2’s. Y axis negative 8 to 16, by 4s. Opens upward with vertex at the origin. Parabola labeled y equals open parentheses, x minus 3, closed parenthesis, squared. Opens upward with vertex at 3 comma 0. Parabola labeled y equals open parenthesis x minus 3, closed parenthesis, squared, plus seven. Opens upwards with vertex at 3 comma 7.

2. How does the -3 in \((x-3)^2\) affect the graph when compared to \(y = x^2\)?
3. How does the +7 in \((x-3)^2 + 7 \) affect the graph when compared to \(y = (x-3)^2\)?