Finding the Translated Parabola: y=x² with Root at x=4

Q: Why does $ y = (x - 4)^2 $ have its root at x = 4?

When x = 4, we get $ y = (4 - 4)^2 = 0^2 = 0 $. For any other x-value, like x = 3, we get $ y = (3 - 4)^2 = (-1)^2 = 1 > 0 $.

Q: How do I remember which direction the parabola moves?

Think of it as "opposite day"! In $ y = (x - h)^2 $, if you want the vertex at x = 4, you need h = 4, so it becomes $ (x - 4)^2 $. The minus sign might seem backwards, but it works!

Q: What does 'positive in all areas except x = 4' mean exactly?

This means the parabola is above the x-axis (y > 0) everywhere except at exactly x = 4, where it touches the x-axis (y = 0). The parabola never goes below the x-axis.

Q: Could the answer be $ y = (x - 4)^2 + 1 $ instead?

No! Adding 1 moves the vertex up to (4, 1), so the parabola never touches the x-axis. We need the parabola to equal zero at x = 4, which requires the vertex to be at (4, 0).

Q: How is this different from the original $ y = x^2 $ ?

The original $ y = x^2 $ has its vertex at (0, 0) and root at x = 0. Our answer $ y = (x - 4)^2 $ shifts this parabola 4 units right, moving the vertex to (4, 0) and the root to x = 4.

Parabola Translations with Root Conditions

Which parabola is the translation of the graph of the function $y=x^2$

and is positive in all areas except $x=4$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find the right function based on our data.

00:13 We'll use the positive domain, which tells us the parabola opens upwards.

00:18 This means the coefficient for X squared is positive.

00:22 We'll draw the function using the intersection points and parabola type.

00:28 The shift on the X axis depends on the term P.

00:34 We'll plug these values into the parabola formula and solve to find the function.

00:39 And that's how we solve this problem! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which parabola is the translation of the graph of the function $y=x^2$

and is positive in all areas except $x=4$ ?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the desired vertex of the parabola.
Step 2: Use the vertex form of the parabola, $y = (x - h)^2 + k$ .
Step 3: Verify the solution with problem constraints.

Now, let's work through each step:
Step 1: We need the parabola to have a vertex such that it equals zero at $x = 4$ . Thus, the vertex is $(4, 0)$ .
Step 2: Using the vertex form, substitute $h = 4$ and $k = 0$ into $y = (x - h)^2 + k$ , resulting in $y = (x - 4)^2$ . This equation ensures that $y = 0$ only when $x = 4$ .
Step 3: This parabola is positive for values of $x$ other than 4, as the square of any nonzero number is positive. Thus, it meets the specified condition of being positive except at $x = 4$ .

Therefore, the solution to the problem is $y = (x - 4)^2$ .

Final Answer

$y=(x-4)^2$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Use $y = (x - h)^2 + k$ where vertex is (h, k)
Root Translation: For root at x = 4, vertex is (4, 0) giving $y = (x - 4)^2$
Verification: Check that y = 0 only at x = 4 and y > 0 everywhere else ✓

Common Mistakes

Avoid these frequent errors

Confusing the direction of horizontal translation
Don't write $y = (x + 4)^2$ for a root at x = 4 = vertex moves to (-4, 0)! This creates a root at x = -4, not x = 4. Always remember: $y = (x - h)^2$ moves the vertex to (h, 0), so for x = 4, use h = 4.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

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

With the Y

FAQ

Everything you need to know about this question

Why does $y = (x - 4)^2$ have its root at x = 4?

When x = 4, we get $y = (4 - 4)^2 = 0^2 = 0$ . For any other x-value, like x = 3, we get $y = (3 - 4)^2 = (-1)^2 = 1 > 0$ .

How do I remember which direction the parabola moves?

Think of it as "opposite day"! In $y = (x - h)^2$ , if you want the vertex at x = 4, you need h = 4, so it becomes $(x - 4)^2$ . The minus sign might seem backwards, but it works!

What does 'positive in all areas except x = 4' mean exactly?

This means the parabola is above the x-axis (y > 0) everywhere except at exactly x = 4, where it touches the x-axis (y = 0). The parabola never goes below the x-axis.

Could the answer be $y = (x - 4)^2 + 1$ instead?

No! Adding 1 moves the vertex up to (4, 1), so the parabola never touches the x-axis. We need the parabola to equal zero at x = 4, which requires the vertex to be at (4, 0).

How is this different from the original $y = x^2$ ?

The original $y = x^2$ has its vertex at (0, 0) and root at x = 0. Our answer $y = (x - 4)^2$ shifts this parabola 4 units right, moving the vertex to (4, 0) and the root to x = 4.

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Finding the Translated Parabola: y=x² with Root at x=4

Parabola Translations with Root Conditions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does $y = (x - 4)^2$ have its root at x = 4?

How do I remember which direction the parabola moves?

What does 'positive in all areas except x = 4' mean exactly?

Could the answer be $y = (x - 4)^2 + 1$ instead?

How is this different from the original $y = x^2$ ?

🌟 Unlock Your Math Potential

More Questions

Parabola of the Form y=(x-p)²

Match the Graph: Identifying y=x² Among Four Parabola Options

Match the Quadratic Function y=(x+3)² with its Corresponding Graph

Parabola Translation: Finding y=x² Shift with Exception at x=-3

Transforming y=-x^2: Finding the Parabola Negative Everywhere Except at x=-4

Finding the Translated Parabola: y=x² with Root at x=4

Parabola Translations with Root Conditions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does y=(x−4)2 y = (x - 4)^2 y=(x−4)2 have its root at x = 4?

How do I remember which direction the parabola moves?

What does 'positive in all areas except x = 4' mean exactly?

Could the answer be y=(x−4)2+1 y = (x - 4)^2 + 1 y=(x−4)2+1 instead?

How is this different from the original y=x2 y = x^2 y=x2?

🌟 Unlock Your Math Potential

More Questions

Parabola of the Form y=(x-p)²

Match the Graph: Identifying y=x² Among Four Parabola Options

Match the Quadratic Function y=(x+3)² with its Corresponding Graph

Parabola Translation: Finding y=x² Shift with Exception at x=-3

Transforming y=-x^2: Finding the Parabola Negative Everywhere Except at x=-4

Why does $y = (x - 4)^2$ have its root at x = 4?

Could the answer be $y = (x - 4)^2 + 1$ instead?

How is this different from the original $y = x^2$ ?