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

Family of Parabolas $y=(x-p)²+k$

Combination of Horizontal and Vertical Shift
In this quadratic function
$K$ 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.

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:

1 - combination of horizontal and vertical shift

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.

Join Over 30,000 Students Excelling in Math!

Endless Practice, Expert Guidance - Elevate Your Math Skills Today

Test your knowledge

Question 1

Find the corresponding algebraic representation of the drawing:

Question 2

Find the corresponding algebraic representation of the drawing:

Question 3

Choose the equation that represents the function

$$ y=-x^2 $$

moved 3 spaces to the left

and 4 spaces up.

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?

Question 1

Which equation represents the function:

$$ y=x^2 $$

moved 2 spaces to the right

and 5 spaces upwards.

Question 2

Choose the equation that corresponds to the function

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

moved 3 spaces up.

Question 3

Which equation represents

$$ y=(x-1)^2 $$

moved units spaces downward?

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

Exercise #1

Find the corresponding algebraic representation of the drawing:

Video Solution

Answer

$y=x^2-4$

Exercise #2

Find the corresponding algebraic representation of the drawing:

Video Solution

Answer

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

Exercise #3

Find the corresponding algebraic representation of the drawing:

Video Solution

Answer

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

Exercise #4

Choose the equation that represents the function

$y=-x^2$

moved 3 spaces to the left

and 4 spaces up.

Video Solution

Answer

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

Exercise #5

Which equation represents the function:

$y=x^2$

moved 2 spaces to the right

and 5 spaces upwards.

Video Solution

Answer

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

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

Family of Parabolas y=(x−p)2+k y=(x-p)²+k y=(x−p)2+k

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

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

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

Algebraic Solution

Graphical Solution

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

Exercise #1

Video Solution

Answer

Exercise #2

Video Solution

Answer

Exercise #3

Video Solution

Answer

Exercise #4

Video Solution

Answer

Exercise #5

Video Solution

Answer

Family of Parabolas $y=(x-p)²+k$

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