Family of Parabolas y=(x-p)²

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

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

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

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

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

1 - The function   Y=(X+2)^2

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


Find the intersection of the function

\( y=(x+4)^2 \)

With the Y

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Change in Slope

When there is an aa before the parentheses, it indicates the slope of the parabola.
The larger the aa, the closer the parabola will be to its axis of symmetry. Steeper - with a smaller opening.
The smaller the aa, the farther the parabola will be from its axis of symmetry. Less steep - with a larger opening.

Let's see an example that shows the balance between the change in slope and horizontal shift:
For example, in the function 

Y=12(X2)2Y=\frac{1}{2} (X-2)^2

the changes will be:
Shift of 22 steps to the right
and the change in slope, The parabola will be less steep and with a larger opening
Let's see it in an illustration:

2 - Shift of 2 steps to the right

Graphical and Algebraic Solution when y=0y=0

Algebraic Solution

when y=0y=0 the expression inside parentheses must be equal to 0 0 .
XX must be equal to PP for the equation to be correct.

Graphical Solution

The graphical solution is the vertex of the parabola.
In a quadratic function of this form 
in which the coefficient of X2X^2  is 11,
the vertex of the parabola is composed ofY=0 Y=0  and byPP which indicates the XX vertex.
Note that, when in the equation there is a minus sign before thePP
indeed, it is positive and, when there is a plus sign before the PP it is, in fact, negative.

Examples and exercises with solutions from the family of parabolas y=(x-p)²


Find the positive domain of the function

y=(x2)2 y=(x-2)^2


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

0 = (x-2)²


We perform a square root:



And thus we reveal the point

(2, 0)

This is the extreme point of the parabola.


Then we decompose the equation into standard form:




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 extreme point,

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



all x, x2 x\ne2

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