Parabola Families - Examples, Exercises and Solutions

Practice Families of ParabolasPractice The functions y=x²Practice Family of Parabolas y=x²+c: Vertical ShiftPractice Family of Parabolas y=(x-p)²Practice Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)
Question Types:
Parabola of the form y=x²: Write a function that expresses..Parabola of the Form y=x²+c: Linking function properties to its representationParabola of the form y=x²: Complete the missing numbersParabola of the form y=x²: Determine whether the function is defined at a certain pointParabola of the Form y=x²+c: Finding Increasing or Decreasing DomainsParabola of the Form y=x²+c: Identify positive and negative domains on a graphParabola of the Form y=x²+c: Match functions to the appropriate graphsParabola of the Form y=(x-p)²: Match functions to the appropriate graphsParabola of the Form y=(x-p)²: True / falseParabola of the Form y=(x-p)²: Determine the algebraic representation of the following descriptionsParabola of the Form y=(x-p)²: Identify the positive and negative domainParabola of the Form y=(x-p)²+k: Identify the positive and negative domainParabola of the form y=x²: Finding a stationary pointParabola of the Form y=x²+c: Finding points on the graph of a functionParabola of the Form y=(x-p)²: Finding a stationary pointParabola of the Form y=(x-p)²: Finding Increasing or Decreasing DomainsParabola of the Form y=(x-p)²+k: Finding a stationary pointParabola of the Form y=(x-p)²+k: Finding Increasing or Decreasing DomainsParabola of the Form y=(x-p)²+k: Determine the algebraic representation of the following descriptions

The Family of Parabolas

The function \(y=x^2\)

the most basic quadratic function:
y=X2y=X^2

Parabola y=X²

The family of parabolas \(y=x²+c\)

The family of parabolas y=x2+cy=x^2+c
The basic quadratic function – with the addition of cc

The family of parabolas \(y=(x-p)²\)

In this family, we are given a quadratic function that clearly shows us how the function moves horizontally – how many steps it needs to move right or left.
PP represents the number of steps the function will move horizontally – right or left.
If PP is positive – (there is a minus in the equation) – the function will move PP steps to the right.
If PP is negative – (and as a result, there is a plus in the equation because minus times minus equals plus) – the function will move PP steps to the left.

The family of parabolas \(y=(x-p)²+k\)

In this quadratic function, we can see a combination of horizontal and vertical shifts:
KK: Determines the number of steps and the direction the function will move vertically – up or down.
KK positive – shift up, KK negative – shift down.
PP: Determines the number of steps and the direction the function will move horizontally – right or left.

Detailed explanation

Practice Parabola Families

Question 1

What is the positive domain of the function below?

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

Question 2

Find the intersection of the function

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

With the Y

Question 3

Find the intersection of the function

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

With the X

Question 4

What is the value of y for the function?

\( y=x^2 \)

of the point \( x=2 \)?

Question 5

Complete:

The missing value of the function point:

\( f(x)=x^2 \)

\( f(?)=16 \)

Examples with solutions for Parabola Families

Exercise #1

What is the positive domain of the function below?

y=(x−2)2 y=(x-2)^2

Video Solution

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, x≠2 x\ne2

Exercise #2

Find the intersection of the function

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

With the Y

Video Solution

Answer

(0,16) (0,16)

Exercise #3

Find the intersection of the function

y=(x−2)2 y=(x-2)^2

With the X

Video Solution

Answer

(2,0) (2,0)

Exercise #4

What is the value of y for the function?

y=x2 y=x^2

of the point x=2 x=2 ?

Video Solution

Answer

y=4 y=4

Exercise #5

Complete:

The missing value of the function point:

f(x)=x2 f(x)=x^2

f(?)=16 f(?)=16

Video Solution

Answer

f(4) f(4) f(−4) f(-4)

Question 1

Find the ascending area of the function

\( f(x)=2x^2 \)

Question 2

Find the descending area of the function

\( f(x)=\frac{1}{2}x^2 \)

Question 3

Which chart represents the function \( y=x^2-9 \)?

222333999-9-9-9-1-1-1444-101234

Question 4

One function

\( y=6x^2 \)

to the corresponding graph:

1234

Question 5

One function

\( y=-6x^2 \)

to the corresponding graph:

1234

Exercise #6

Find the ascending area of the function

f(x)=2x2 f(x)=2x^2

Video Solution

Answer

0 < x

Exercise #7

Find the descending area of the function

f(x)=12x2 f(x)=\frac{1}{2}x^2

Video Solution

Answer

x < 0

Exercise #8

Which chart represents the function y=x2−9 y=x^2-9 ?

222333999-9-9-9-1-1-1444-101234

Video Solution

Answer

4

Exercise #9

One function

y=6x2 y=6x^2

to the corresponding graph:

1234

Video Solution

Answer

2

Exercise #10

One function

y=−6x2 y=-6x^2

to the corresponding graph:

1234

Video Solution

Answer

4

Question 1

One function

\( y=-2x^2-3 \)

to the corresponding graph:

333333-3-3-3333-3-3-3-3-3-31234

Question 2

Find the intersection of the function

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

With the Y

Question 3

Find the intersection of the function

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

With the Y

Question 4

Find the positive area of the function

\( y=(x+6)^2 \)

Question 5

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

Exercise #11

One function

y=−2x2−3 y=-2x^2-3

to the corresponding graph:

333333-3-3-3333-3-3-3-3-3-31234

Video Solution

Answer

4

Exercise #12

Find the intersection of the function

y=(x−2)2 y=(x-2)^2

With the Y

Video Solution

Answer

(0,4) (0,4)

Exercise #13

Find the intersection of the function

y=(x−6)2 y=(x-6)^2

With the Y

Video Solution

Answer

(0,36) (0,36)

Exercise #14

Find the positive area of the function

y=(x+6)2 y=(x+6)^2

Video Solution

Answer

x≠−6 x\ne-6

Exercise #15

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

Video Solution

Answer

For each X x≠5 x\ne5