Determine the Intersection Points: Graphing y = (x-1)(x-1)

Q: Why is there only one x-intercept instead of two?

Because $ y = (x-1)^2 $ is a perfect square! When you have $ (x-1)^2 = 0 $, the only way this can be zero is if x-1 = 0, giving us x = 1. This is called a repeated root or double root.

Q: What does this look like on a graph?

The parabola touches the x-axis at (1,0) but doesn't cross through it. It's like a ball bouncing off the ground - it touches at one point and goes back up. This is different from a parabola that crosses the x-axis at two points.

Q: How is this different from y = (x-1)(x+1)?

Great comparison! $ y = (x-1)(x+1) $ has two different factors, so it crosses the x-axis at two points: (1,0) and (-1,0). But $ y = (x-1)^2 $ has the same factor twice, creating only one x-intercept.

Q: Can I expand (x-1)² first?

Yes! $ (x-1)^2 = x^2 - 2x + 1 $. Then set $ x^2 - 2x + 1 = 0 $. You can use the quadratic formula or factor back to $ (x-1)^2 = 0 $. Either way, you get x = 1.

Q: What if the question asked for y-intercept instead?

For the y-intercept, set x = 0: $ y = (0-1)^2 = (-1)^2 = 1 $. So the y-intercept would be (0,1). Remember: x-intercept means y = 0, y-intercept means x = 0!

Quadratic Functions with Repeated Roots

Determine the points of intersection of the function

$y=(x-1)(x-1)$

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 points with the X-axis

00:03 At the intersection points with the X-axis, the Y value must = 0

00:07 Substitute Y = 0 and solve to find X values

00:13 Find what makes each factor in the product zero

00:16 This is one solution

00:21 This solution makes both factors in the product zero

00:24 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

Determine the points of intersection of the function

$y=(x-1)(x-1)$

With the X

Step-by-step solution

To solve this problem, we'll determine the intersection points of the function $y = (x - 1)(x - 1)$ with the x-axis by following these steps:

Step 1: Set the function equal to zero to find where it intersects the x-axis.
Step 2: Solve the equation $(x - 1)^2 = 0$ .
Step 3: Determine the x-coordinate(s) from this equation.
Step 4: The y-coordinate will be zero at these points.

Let's work through these steps:

Step 1: We set the given function to zero: $y = (x - 1)^2 = 0$ .

Step 2: By solving the equation $(x - 1)^2 = 0$ , we apply the property that a square is zero only if the base is zero.

Step 3: Solving $(x - 1) = 0$ , we find:

$x = 1$

Step 4: The corresponding point on the graph is $(1, 0)$ , indicating where the function crosses the x-axis.

Therefore, the point of intersection of the function with the x-axis is $(1, 0)$ .

Final Answer

$(1,0)$

Key Points to Remember

Essential concepts to master this topic

X-intercepts: Set y = 0 and solve for x values
Technique: $(x-1)^2 = 0$ gives x - 1 = 0, so x = 1
Check: Substitute x = 1: $(1-1)^2 = 0^2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Finding two different x-intercepts for squared factors
Don't assume $(x-1)^2 = 0$ gives two different roots like $(1,0)$ and $(-1,0)$ ! A squared factor creates a repeated root, meaning the parabola touches the x-axis at only one point. Always recognize that $(x-a)^2 = 0$ has exactly one solution: x = a.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

$$ f(x)=-x^2+5x+6 $$

Calculate points A and B.

FAQ

Everything you need to know about this question

Why is there only one x-intercept instead of two?

Because $y = (x-1)^2$ is a perfect square! When you have $(x-1)^2 = 0$ , the only way this can be zero is if x-1 = 0, giving us x = 1. This is called a repeated root or double root.

What does this look like on a graph?

The parabola touches the x-axis at (1,0) but doesn't cross through it. It's like a ball bouncing off the ground - it touches at one point and goes back up. This is different from a parabola that crosses the x-axis at two points.

How is this different from y = (x-1)(x+1)?

Great comparison! $y = (x-1)(x+1)$ has two different factors, so it crosses the x-axis at two points: (1,0) and (-1,0). But $y = (x-1)^2$ has the same factor twice, creating only one x-intercept.

Can I expand (x-1)² first?

Yes! $(x-1)^2 = x^2 - 2x + 1$ . Then set $x^2 - 2x + 1 = 0$ . You can use the quadratic formula or factor back to $(x-1)^2 = 0$ . Either way, you get x = 1.

What if the question asked for y-intercept instead?

For the y-intercept, set x = 0: $y = (0-1)^2 = (-1)^2 = 1$ . So the y-intercept would be (0,1). Remember: x-intercept means y = 0, y-intercept means x = 0!

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Determine the Intersection Points: Graphing y = (x-1)(x-1)

Quadratic Functions with Repeated Roots

❤️ 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 instead of two?

What does this look like on a graph?

How is this different from y = (x-1)(x+1)?

Can I expand (x-1)² first?

What if the question asked for y-intercept instead?

🌟 Unlock Your Math Potential

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