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 function without using graphing technology. Make sure the \(x\)-intercepts, \(y\)-intercept, and vertex are plotted accurately.

Here is one way an expression in standard form is rewritten into vertex form.

\(\begin{align}&x^2 - 7x + 6 &\qquad &\text{original expression}\\ &x^2 - 7x + \left(\text-\frac72\right)^2 + 6 -\left(\text- \frac72\right)^2 &\quad&\text{step 1} \\ &\left(x-\frac72\right)^2 + 6-\frac{49}{4} &\quad&\text{step 2}\\ &\left(x-\frac72\right)^2 + \frac{24}{4}-\frac{49}{4} &\quad&\text{step 3}\\ &\left(x-\frac72\right)^2-\frac{25}{4} &\quad&\text{step 4} \end{align}\)

1. In step 1, where does the number \(\text-\frac72\) come from?
2. In step 1, why is \(\left(\text-\frac72\right)^2\) added and then subtracted?
3. What happens in step 2?
4. What happens in step 3?
5. What does the last expression tell us about the graph of a function defined by this expression?

Rewrite each quadratic expression in vertex form.

1. \(d(x) = x^2 + 12x + 36\)
2. \(f(x) = x^2 + 10x + 21\)
3. \(g(x) = 2x^2 - 20x + 32\)

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

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

Select all the equations with irrational solutions.

1. What are the coordinates of the vertex of the graph of the function defined by \(f(x)=2(x+1)^2-4\)?
2. Find the coordinates of two other points on the graph.

3. Sketch the graph of \(f\).

   

How is the graph of the equation \(y=(x-1)^2 + 4\) related to the graph of the equation \(y=x^2\)?