Parabola of the Form y=(x-p)² - Examples, Exercises and Solutions

Understanding Parabola of the Form y=(x-p)²

Complete explanation with examples

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


Detailed explanation

Practice Parabola of the Form y=(x-p)²

Test your knowledge with 11 quizzes

One function

\( y=-x^2 \)

for the corresponding chart

-1-1-11234

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

Step-by-step solutions included
Exercise #1

Find the intersection of the function

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

With the Y

Step-by-Step Solution

To solve this problem, we will find the intersection of the function with the Y-axis by following these steps:

  • Step 1: Recognize that the intersection with the Y-axis occurs where x=0 x = 0 .
  • Step 2: Substitute x=0 x = 0 into the function y=(x+4)2 y = (x+4)^2 .
  • Step 3: Perform the calculation to find the y-coordinate.

Now, let's solve the problem:

Step 1: Identify the Y-axis intersection by setting x=0 x = 0 .
Step 2: Substitute x=0 x = 0 into the function:

y=(0+4)2=42=16 y = (0+4)^2 = 4^2 = 16

Step 3: The intersection point on the Y-axis is (0,16)(0, 16).

Therefore, the solution to the problem is (0,16)(0, 16).

Answer:

(0,16) (0,16)

Video Solution
Exercise #2

Find the intersection of the function

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

With the X

Step-by-Step Solution

To solve this problem, we'll find the intersection of the function y=(x2)2 y = (x-2)^2 with the x-axis. The x-axis is characterized by y=0 y = 0 . Hence, we set (x2)2=0 (x-2)^2 = 0 and solve for x x .

Let's follow these steps:

  • Step 1: Set the function equal to zero:

(x2)2=0 (x-2)^2 = 0

  • Step 2: Solve the equation for x x :

Taking the square root of both sides gives x2=0 x - 2 = 0 .

Adding 2 to both sides results in x=2 x = 2 .

  • Step 3: Find the intersection point coordinates:

The x-coordinate is x=2 x = 2 , and since it intersects the x-axis, the y-coordinate is y=0 y = 0 .

Therefore, the intersection point of the function with the x-axis is (2,0)(2, 0).

The correct choice from the provided options is (2,0) (2, 0) .

Answer:

(2,0) (2,0)

Video Solution
Exercise #3

Find the intersection of the function

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

With the Y

Step-by-Step Solution

To determine the intersection of the function y=(x2)2 y = (x-2)^2 with the y-axis, we set x=0 x = 0 , as the y-axis is defined by all points where x=0 x = 0 .

Substituting x=0 x = 0 into the equation:

y=(02)2 y = (0 - 2)^2

Simplifying this expression:

y=(2)2=4 y = (-2)^2 = 4

Thus, the intersection point of the function with the y-axis is (0,4) (0, 4) .

Therefore, the solution to the problem is (0,4) (0, 4) .

Answer:

(0,4) (0,4)

Video Solution
Exercise #4

Find the intersection of the function

y=(x6)2 y=(x-6)^2

With the Y

Step-by-Step Solution

To find the intersection of the function y=(x6)2 y = (x-6)^2 with the y-axis, we follow these steps:

  • Step 1: Identify the known function and approach the problem by setting x=0 x = 0 since we are looking for the intersection with the y-axis.

  • Step 2: Substitute x=0 x = 0 into the equation y=(x6)2 y = (x-6)^2 .

  • Step 3: Perform the calculation to find y y .

Now, execute these steps:
Step 1: We are given the function y=(x6)2 y = (x-6)^2 .
Step 2: Substitute x=0 x = 0 into the equation:
y=(06)2 y = (0-6)^2
Step 3: Simplify the expression:
y=(6)2=36 y = (-6)^2 = 36

The point of intersection with the y-axis is therefore (0,36) (0, 36) .

Thus, the solution to the problem is (0,36) (0, 36) .

Answer:

(0,36) (0,36)

Video Solution
Exercise #5

What is the positive domain of the function below?

y=(x2)2 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.

 

Answer:

all x, x2 x\ne2

Video Solution

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