# Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)

🏆Practice parabola of the form y=(x-p)²+k

### Family of Parabolas$y=(x-p)²+k$

Combination of Horizontal and Vertical Shift
$K$ determines the amount of steps and the vertical direction in which the function will shift - upwards or downwards.
$P$ determines the amount of steps and the horizontal direction in which the function will shift - to the right or to the left.

## Test yourself on parabola of the form y=(x-p)²+k!

Which equation represents the function:

$$y=x^2$$

moved 2 spaces to the right

and 5 spaces upwards.

## Let's look at an example of combining both displacements together.

For example, in the function:
$y=(x-4)^2+3$

The changes will be:
according to $P=4$ -The parabola will shift $4$ steps to the right.
According to $K=3$ -The parabola will shift $3$ steps upwards.

Let's see it in the illustration:

We can see that the vertex of the parabola is:
$(4,3)$

### Zero Point or Root of the Function - Graphical and Algebraic Solution when$Y=0$

The zeros of a function are the intersections with the $X$ axis.

## Algebraic Solution

when $K$ positive - This equation has no solution except in the case where $K$ equals$0$ and also $P$ equals$X$.
when $K$ negative- Generally, this equation will have $2$ solutions.

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## Graphical Solution

The graphical solution are the points of intersection of the parabola with the $X$ axis
that is, the zeros of the function.

Do you know what the answer is?

## examples with solutions for parabola of the form y=(x-p)²+k

### Exercise #1

Which equation represents the function:

$y=x^2$

moved 2 spaces to the right

and 5 spaces upwards.

### Video Solution

$y=(x-2)^2+5$

### Exercise #2

Find the corresponding algebraic representation of the drawing:

### Video Solution

$y=(x+2)^2+7$

### Exercise #3

Find the corresponding algebraic representation of the drawing:

### Video Solution

$y=x^2-4$

### Exercise #4

Choose the equation that represents the function

$y=-x^2$

moved 3 spaces to the left

and 4 spaces up.

### Video Solution

$y=-(x+3)^2+4$

### Exercise #5

Find the corresponding algebraic representation of the drawing:

### Video Solution

$y=(x-5)^2+4$