Investigating $y=(x-h)^2+k$

$y=x^2$ is called a quadratic equation because the highest power of the independent variable, $x$, is 2. In this lab you will study this and other similar quadratic equations.

Question 1. If $y=x^2$, what will the value of $y$ be if $x=3$? Enter this value in the $y$ column corresponding to $x=3$ in the table to the right. Repeat this for all of the other $x$ values in the table, using the equation $y=x^2$.

$x$	$y$
3
2
1	1
0
-1
-2	4
-3

Notice that the points from the table above are plotted on the grid to the left. Click on to graph the equation $y=x^2$ on the same grid.

Can you find any value for $x$ so that when you plug it into $y=x^2$ you get a negative number? If so, give an example. If not, explain why not.

What is the smallest possible value $y$ can have?

Is there a largest possible value $y$ can have?

Question 2. Do you notice any symmetry? For example, can you fold the grid along a line so that the points on one side match the points on the other side? What is the line along which you can fold the grid?

Question 3. Write a paragraph explaining the characteristics of the graph of $y=x^2$. Use ideas that were talked about in the previous questions, as well as any others that might help explain the graph to someone who doesn't know about quadratic equations.

Parabola:The shape of the graph 
(shown above) of $y=x^2$ is called a parabola.
Vertex:In this case, the point $(0,0)$ is
called the vertex of the parabola.
Axis of Symmetry:In this case, the
line $x=0$ is called the axis of symmetry of the parabola.

Equations of the form $y=x^2+k$

$y=x^2+2$ is also a quadratic equation because the highest power of $x$ is 2. In this section you will see how equations like $y=x^2+2$ compare to $y_1=x^2$.

Question 4. Click on to see the next example. Complete this table of values for $y=x^2+2$. The table has been completed for $y_1=x^2$.

$x$	$y_1$	$y$
3	9
2	4	6
1	1
0	0
-1	1
-2	4
-3	9

What is the relationship between the values of $y$ and the values of $y_1$?

Using the table, what is the axis of symmetry of $y=x^2+2$?

Do the blue points seem to be symmetrical around the axis of symmetry?

Question 5. Click on to look at the graphs of $y_1=x^2$ and $y=x^2+2$ together. Compare the two graphs. How are they similar? How are they different? Which graph is higher (that is, which one has bigger $y_1$ or $y$ coordinates)?

Question 6. Click on to see the graph of $y=x^2+k$. Click on the slider for $k$ and move it. As you change $k$, the equation for the graph is written below the grid.

What happens to the graph of $y=x^2+k$ as $k$ increases from 0 to 5?

Where are the vertex and axis of symmetry of $y=x^2+2$?

Vertex
Axis

Where are the vertex and axis of symmetry of $y=x^2+4$?

Vertex
Axis

What is the relationship between the value of $k$ and the vertex of the parabola?

Equations of the form $y=(x-h)^2$

$y=(x-2)^2$ is also a quadratic equation, even though the $x^2$ term is hidden in the way the equation is written. In this section you will learn about equations like $y=(x-2)^2$.

Question 7. Click on to see the example for question 7. Complete the table of values for $y=(x-2)^2$.

$x$	$y_1$	$y$
4	16
3	9
2	4	0
1	1
0	0
-1	1
-2	4
-3	9	25

Looking at the table, you can see that $x=0$ is the axis of symmetry of $y_1=x^2$. What is the axis of symmetry of $y=(x-2)^2$?

How many rows up or down do you need to move the $y_1$ column to match the $y$ column?

Question 8. Click on to look at the graphs of $y_1=x^2$ and $y=(x-2)^2$. Compare the two graphs. How are they similar? How are they different? Which graph's vertex has a bigger $x$-coordinate?

Question 9. Click on to see the example for question 9. Click on the slider for $h$ and move it to change $h$ in the equation $y=(x-h)^2$.

What happens to the graph of $y=(x-h)^2$ as $h$ decreases from 0 to −5?

Use the slider to complete the table below.

Equation	$h$	vertex	axis of symmetry
$y=(x-2)^2$	2
$y=(x-3)^2$

What is the relationship between the value of $h$ and the axis of symmetry?

Putting it all together

Question 10. Click on to see the graph of the equation $y=(x-h)^2+k$. Using the sliders for $h$ and $k$, change their values and notice the effect on the graph. The equation for the parabola is shown below the grid.

Complete the table below.

Equation	$k$	vertex
$y=(x-4)^2+1$	1	$(4,1)$
$y=(x-3)^2-2$
$y=(x+1)^2-5$

Question 11. Where is the vertex of the parabola $y=(x-2.4)^2+1$?

Write the equation for a parabola in the form $y=(x-h)^2+k$, if its vertex is at $(2,3)$.

$y=(\,$$\,)^2+\,$

Question 12. In your own words, describe the general behavior of the graph of $y=(x-h)^2+k$. What effects do $h$ and $k$ have? What is the vertex of this parabola? What is its axis of symmetry? What are the smallest/largest $y$ values it can have?

Parabola:	The shape of the graph (shown above) of $y=x^2$ is called a parabola.
Vertex:	In this case, the point $(0,0)$ is called the vertex of the parabola.
Axis of Symmetry:	In this case, the line $x=0$ is called the axis of symmetry of the parabola.

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