Parabola y=(x-p)² Practice Problems with Solutions

Master parabola transformations y=(x-p)² with step-by-step practice problems. Learn horizontal shifts, vertex identification, and graphing techniques.

📚What You'll Master in This Practice Session

Identify horizontal shifts in parabolas of the form y=(x-p)²
Determine vertex coordinates for transformed parabolas
Graph parabolas with horizontal translations step-by-step
Solve algebraically when y=0 for parabola intersections
Analyze how coefficient 'a' affects parabola steepness and opening
Convert between different forms of quadratic equations

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

Complete explanation with examples

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

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

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

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

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

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

$y=(x+4)^2$

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 $x = 0$ .
Step 2: Substitute $x = 0$ into the function $y = (x+4)^2$ .
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 $x = 0$ .
Step 2: Substitute $x = 0$ into the function:

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

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

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

Answer:

$(0,16)$

Video Solution

Exercise #2

Find the intersection of the function

$y=(x-2)^2$

With the X

Step-by-Step Solution

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

Let's follow these steps:

Step 1: Set the function equal to zero:

$(x-2)^2 = 0$

Step 2: Solve the equation for $x$ :

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

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

Step 3: Find the intersection point coordinates:

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

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

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

Answer:

$(2,0)$

Video Solution

Exercise #3

Find the intersection of the function

$y=(x-2)^2$

With the Y

Step-by-Step Solution

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

Substituting $x = 0$ into the equation:

$y = (0 - 2)^2$

Simplifying this expression:

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

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

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

Answer:

$(0,4)$

Video Solution

Exercise #4

Find the intersection of the function

$y=(x-6)^2$

With the Y

Step-by-Step Solution

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

Step 1: Identify the known function and approach the problem by setting $x = 0$ since we are looking for the intersection with the y-axis.
Step 2: Substitute $x = 0$ into the equation $y = (x-6)^2$ .
Step 3: Perform the calculation to find $y$ .

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

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

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

Answer:

$(0,36)$

Video Solution

Exercise #5

What is the positive domain of the function below?

$y=(x-2)^2$

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, $x\ne2$

Video Solution

Frequently Asked Questions

Everything you need to know about Parabola of the Form y=(x-p)²

How do you find the vertex of y=(x-p)² parabolas?

The vertex of y=(x-p)² is at point (p, 0). If the equation is y=(x-3)², the vertex is (3, 0). If it's y=(x+2)², rewrite as y=(x-(-2))², so the vertex is (-2, 0).

What is the difference between y=(x-p)² and y=(x+p)² transformations?

• y=(x-p)² shifts the parabola p units to the RIGHT • y=(x+p)² shifts the parabola p units to the LEFT • The sign in the parentheses is opposite to the direction of movement

How does the coefficient 'a' in y=a(x-p)² affect the parabola?

The coefficient 'a' controls the parabola's steepness and opening width. Larger values of |a| make the parabola steeper with a narrower opening. Smaller values of |a| make it wider with a gentler curve.

When does y=(x-p)² equal zero and what does this mean graphically?

y=(x-p)² equals zero when x=p. Graphically, this represents the x-intercept where the parabola touches the x-axis, which is also the x-coordinate of the vertex.

What are the steps to graph y=(x-p)² parabolas?

1. Identify the value of p to find horizontal shift 2. Plot the vertex at (p, 0) 3. Choose x-values on both sides of the vertex 4. Calculate corresponding y-values using the equation 5. Connect points to form the parabola shape

How do you solve word problems involving y=(x-p)² parabolas?

First, identify what the variables represent in context. Then determine the vertex from the given information, write the equation in y=(x-p)² form, and use algebraic or graphical methods to find the required values.

What is the axis of symmetry for parabolas in the form y=(x-p)²?

The axis of symmetry is the vertical line x=p. This line passes through the vertex and divides the parabola into two mirror-image halves.

Can y=(x-p)² parabolas open downward?

The basic form y=(x-p)² always opens upward. To open downward, you need a negative coefficient: y=-a(x-p)² where a is positive. This creates a parabola that opens downward with vertex at (p, 0).

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

Click on any question to see the complete solution with step-by-step explanations

Parabola of the Form y=(x-p)²

Continue Your Math Journey

Master Parabola of the Form y=(x-p)² first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

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

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Determine the algebraic representation of the following descriptions Finding a stationary point Finding Increasing or Decreasing Domains Identify the positive and negative domain Match functions to the appropriate graphs True / false

Parabola y=(x-p)² Practice Problems with Solutions

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

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

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

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

Frequently Asked Questions

How do you find the vertex of y=(x-p)² parabolas?

What is the difference between y=(x-p)² and y=(x+p)² transformations?

How does the coefficient 'a' in y=a(x-p)² affect the parabola?

When does y=(x-p)² equal zero and what does this mean graphically?

What are the steps to graph y=(x-p)² parabolas?

How do you solve word problems involving y=(x-p)² parabolas?

What is the axis of symmetry for parabolas in the form y=(x-p)²?

Can y=(x-p)² parabolas open downward?

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

Parabola of the Form y=(x-p)²

Continue Your Math Journey

Suggested Topics to Practice in Advance

Topics Learned in Later Sections

Practice by Question Type

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