### Family of Parabolas $y=(x-p)^2$

In this family, we have a slightly different quadratic function that shows us, very clearly, how the parabola shifts horizontally.
$P$ indicates the number of steps the parabola will move horizontally, to the right or to the left.
If $P$ is positive: (there is a minus sign in the equation) - The parabola will move $P$ steps to the right.
If $P$ is negative: (and, consequently, there will be a plus sign in the equation since minus by minus equals plus) - The parabola will move $P$ steps to the left.

Let's see an example:
The function  $Y=(X+2)^2$

shifts two steps to the left.
Let's see it in an illustration:

## Examples with solutions for Parabola of the Form y=(x-p)²

### Exercise #1

What is the positive domain of the function below?

$y=(x-2)^2$

### Step-by-Step Solution

In the first step, we place 0 in place of Y:

0 = (x-2)²

We perform a square root:

0=x-2

x=2

And thus we reveal the point

(2, 0)

This is the vertex of the parabola.

Then we decompose the equation into standard form:

y=(x-2)²

y=x²-4x+2

Since the coefficient of x² is positive, we learn that the parabola is a minimum parabola (smiling).

If we plot the parabola, it seems that it is actually positive except for its vertex.

Therefore the domain of positivity is all X, except X≠2.

all x, $x\ne2$

### Exercise #2

Find the intersection of the function

$y=(x-2)^2$

With the X

### Video Solution

$(2,0)$

### Exercise #3

Find the intersection of the function

$y=(x+4)^2$

With the Y

### Video Solution

$(0,16)$

### Exercise #4

To work out the points of intersection with the X axis, you must substitute $x=0$.

False

### Exercise #5

To find the y axis intercept, you substitute $x=0$ into the equation and solve for y.

True

### Exercise #6

To which chart does the function $y=x^2$ correspond?

2

### Exercise #7

One function

$y=-x^2$

for the corresponding chart

2

### Exercise #8

Find the positive area of the function

$y=(x+6)^2$

### Video Solution

$x\ne-6$

### Exercise #9

Find the intersection of the function

$y=(x-6)^2$

With the Y

### Video Solution

$(0,36)$

### Exercise #10

Find the negative area of the function

$y=(x+2)^2$

There is no

### Exercise #11

Find the ascending area of the function

$y=(x-3)^2$

3 < x

### Exercise #12

Find the intersection of the function

$y=(x-2)^2$

With the Y

### Video Solution

$(0,4)$

### Exercise #13

Find the descending area of the function

$y=(x+4)^2$

x < -4

### Exercise #14

Find the descending area of the function

$y=(x-5)^2$

x < 5

### Exercise #15

Find the positive area of the function
$y=(x+5)^2$

### Video Solution

For each X $x\ne5$