Find X-Intercepts of y=(x-2)² : Quadratic Function Analysis

Q: Why is there only one x-intercept for this parabola?

Because $ y = (x-2)^2 $ is a perfect square, it only touches the x-axis at one point. The parabola has its vertex right on the x-axis at (2, 0), so this is called a double root.

Q: How is this different from finding y-intercepts?

For x-intercepts, set y = 0 and solve for x. For y-intercepts, set x = 0 and solve for y. They're opposite processes! This function's y-intercept would be at (0, 4).

Q: What if the equation was (x-2)² = 4 instead?

Then you'd have two x-intercepts! Take the square root: x - 2 = ±2, so x = 4 or x = 0. The intercepts would be (0, 0) and (4, 0).

Q: Why do we write the answer as (2, 0) and not just x = 2?

Because an intercept is a point, not just a number! You need both coordinates: the x-value where it crosses (2) and the y-value at that crossing (0).

Q: Can a parabola have no x-intercepts?

Yes! If the parabola opens upward and its vertex is above the x-axis, like $ y = (x-2)^2 + 3 $, it never touches the x-axis and has no real x-intercepts.

X-Intercepts with Perfect Square Forms

Find the intersection of the function

$y=(x-2)^2$

With the X

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the intersection point of the function with the X-axis

00:03 At the intersection point with X-axis, Y=0

00:07 Therefore, we'll set Y=0 and solve to find the intersection point with X-axis

00:13 Take the root to eliminate the power

00:29 Isolate X

00:33 This is the X value at the intersection point, we'll substitute Y=0 as we established at the point

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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 $(2, 0)$ .

Final Answer

$(2,0)$

Key Points to Remember

Essential concepts to master this topic

Definition: X-intercepts occur where the graph crosses the x-axis (y = 0)
Method: Set $(x-2)^2 = 0$ and solve: x - 2 = 0, so x = 2
Check: Substitute x = 2: $(2-2)^2 = 0^2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Confusing x-intercepts with y-intercepts
Don't look for points where x = 0 when finding x-intercepts = you'll get (0, 4) instead of (2, 0)! X-intercepts happen where y = 0, not where x = 0. Always set the function equal to zero and solve for x.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

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

With the X

FAQ

Everything you need to know about this question

Why is there only one x-intercept for this parabola?

Because $y = (x-2)^2$ is a perfect square, it only touches the x-axis at one point. The parabola has its vertex right on the x-axis at (2, 0), so this is called a double root.

How is this different from finding y-intercepts?

For x-intercepts, set y = 0 and solve for x. For y-intercepts, set x = 0 and solve for y. They're opposite processes! This function's y-intercept would be at (0, 4).

What if the equation was (x-2)² = 4 instead?

Then you'd have two x-intercepts! Take the square root: x - 2 = ±2, so x = 4 or x = 0. The intercepts would be (0, 0) and (4, 0).

Why do we write the answer as (2, 0) and not just x = 2?

Because an intercept is a point, not just a number! You need both coordinates: the x-value where it crosses (2) and the y-value at that crossing (0).

Can a parabola have no x-intercepts?

Yes! If the parabola opens upward and its vertex is above the x-axis, like $y = (x-2)^2 + 3$ , it never touches the x-axis and has no real x-intercepts.

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Find X-Intercepts of y=(x-2)² : Quadratic Function Analysis

X-Intercepts with Perfect Square Forms

❤️ 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 is there only one x-intercept for this parabola?

How is this different from finding y-intercepts?

What if the equation was (x-2)² = 4 instead?

Why do we write the answer as (2, 0) and not just x = 2?

Can a parabola have no x-intercepts?

🌟 Unlock Your Math Potential

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