Quadratic Function y=(x-4)²: Analyzing a 4-Unit Horizontal Shift

Q: How can I remember the direction of horizontal shifts?

Use this trick: "Horizontal shifts are backwards!" $ (x - h) $ goes right, $ (x + h) $ goes left. It's opposite to what the sign suggests.

Q: What stays the same when I shift horizontally?

The shape and size of the parabola stay exactly the same! Only the position changes. The vertex moves, but the parabola still opens upward with the same width.

Q: How do I find the new vertex after shifting?

For $ y = (x - h)^2 + k $, the vertex is at (h, k). In our case, $ y = (x - 4)^2 $ has vertex at (4, 0).

Q: Can I verify the shift by checking specific points?

Yes! Pick a point from $ y = x^2 $ like (2, 4). After shifting right 4 units, this becomes (6, 4). Check: $ y = (6-4)^2 = 2^2 = 4 $ ✓

Quadratic Transformations with Horizontal Shifts

$y=(x-4)^2$ is the displacement function $y=x^2$ right 4 steps

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is this function the same as the function shifted 4 steps right?

00:06 Let's start by drawing the markings - we'll note the position 4 steps right

00:15 Let's draw the desired graph

00:19 Let's identify its intersection point with the X-axis

00:22 Let's substitute this point in the proposed function and check if it's correct

00:25 Let's substitute appropriate values and solve

00:28 The equation is correct, meaning the function is correct

00:32 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

$y=(x-4)^2$ is the displacement function $y=x^2$ right 4 steps

Step-by-step solution

To solve this problem, we'll verify if $y = (x-4)^2$ is obtained by shifting $y = x^2$ four units to the right:

Step 1: Identify transformation properties.
The equation $y = (x-h)^2$ indicates a horizontal shift of the parent function $y = x^2$ .
Step 2: Determine the shift value from the expression.
In $y = (x-4)^2$ , $h = 4$ . This implies a shift 4 units to the right because $h = 4$ is positive.
Step 3: Verify the shift.
If we replaced $x$ with $x-4$ , this represents taking the graph of $y = x^2$ and shifting it rightwards by 4 units on the x-axis.

Therefore, we conclude that the function $y = (x-4)^2$ does represent a displacement of $y = x^2$ to the right by 4 units. Hence, the correct answer to the problem is True.

Final Answer

True

Key Points to Remember

Essential concepts to master this topic

Rule: In $y = (x - h)^2$ , positive h shifts right h units
Technique: For $y = (x - 4)^2$ , subtract 4 means shift right 4
Check: Vertex moves from (0,0) to (4,0) confirming rightward shift ✓

Common Mistakes

Avoid these frequent errors

Confusing direction of horizontal shifts
Don't think (x - 4) shifts left because of the minus sign = opposite direction! The minus sign is misleading because we replace x with (x - 4). Always remember: (x - h) shifts RIGHT h units, (x + h) shifts LEFT h units.

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 does (x - 4) shift right when there's a minus sign?

Think of it this way: to get the same y-value as the original function, you need x to be 4 units larger. For example, the original vertex at x = 0 now occurs at x = 4, so the graph moves right!

How can I remember the direction of horizontal shifts?

Use this trick: "Horizontal shifts are backwards!" $(x - h)$ goes right, $(x + h)$ goes left. It's opposite to what the sign suggests.

What stays the same when I shift horizontally?

The shape and size of the parabola stay exactly the same! Only the position changes. The vertex moves, but the parabola still opens upward with the same width.

How do I find the new vertex after shifting?

For $y = (x - h)^2 + k$ , the vertex is at (h, k). In our case, $y = (x - 4)^2$ has vertex at (4, 0).

Can I verify the shift by checking specific points?

Yes! Pick a point from $y = x^2$ like (2, 4). After shifting right 4 units, this becomes (6, 4). Check: $y = (6-4)^2 = 2^2 = 4$ ✓

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