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

Q: Why does (x + 3)² shift the parabola to the LEFT when it has a plus sign?

This confuses many students! Think of it this way: (x + 3)² = 0 when x = -3. The parabola reaches its minimum (vertex) where the expression inside equals zero. So x + 3 = 0 means x = -3, placing the vertex at x = -3.

Q: How do I remember which direction horizontal shifts go?

Use the "opposite rule": The sign inside the parentheses is opposite to the shift direction. (x + 3) shifts LEFT 3 units, (x - 3) shifts RIGHT 3 units. Always solve for when the expression equals zero!

Q: What does 'positive in all areas except x = -3' really mean?

This means the parabola touches the x-axis at exactly one point: x = -3. Everywhere else, the y-values are above the x-axis (positive). The vertex must be at (-3, 0) with the parabola opening upward.

Q: Could the answer be y = -(x + 3)² since it's also zero at x = -3?

No! While y = -(x + 3)² is zero at x = -3, it's negative everywhere else (opens downward). We need the parabola to be positive except at x = -3, so it must open upward.

Q: How can I quickly check my answer?

Test three points: x = -3 (should give y = 0), x = -2 (should be positive), and x = -4 (should be positive). For y = (x + 3)²: at x = -2, y = 1 ✓; at x = -4, y = 1 ✓

Parabola Translation with Vertex Shifts

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

and is positive in all areas except $x=-3$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the correct function based on the data we have.

00:12 We'll use the positive side and see that the parabola smiles.

00:16 This means the coefficient for X squared is positive.

00:21 We'll sketch the function using intersection points and the parabola type.

00:26 The shift to the left depends on the value of the P term.

00:35 Let's substitute into the parabola formula and solve for the function.

00:43 Remember, negative times negative equals positive.

00:49 And that's how we find the solution to our problem.

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 $x=-3$ ?

Step-by-step solution

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

Step 1: Understand the necessary translation of the parabola.
Step 2: Apply the translation to the equation of the parabola.

Now, let's work through each step:

Step 1: We are given the function $y = x^2$ . The new parabola must be zero at $x = -3$ and positive everywhere else. This means the vertex of the parabola is at $(-3, 0)$ .

Step 2: To move the vertex of $y = x^2$ from $(0, 0)$ to $(-3, 0)$ , we need a horizontal shift to the left by 3 units. The translation is represented by replacing $x$ with $x + 3$ in the function:

$y = (x + 3)^2$

This new equation reflects a parabola that opens upwards and has its vertex at $(-3, 0)$ , which means it is zero only at $x = -3$ and positive everywhere else.

Therefore, the solution to the problem is $y = (x + 3)^2$ .

Final Answer

$y=(x+3)^2$

Key Points to Remember

Essential concepts to master this topic

Translation Rule: Replace x with (x + h) to shift left h units
Technique: To shift from (0,0) to (-3,0), use y = (x + 3)²
Check: At x = -3: y = (-3 + 3)² = 0, elsewhere y > 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal shift direction with sign
Don't think y = (x - 3)² shifts left to x = -3! This actually shifts right to x = 3, making the vertex at (3,0) instead. Always remember: (x + h) shifts LEFT h units, (x - h) shifts RIGHT h units.

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 (x + 3)² shift the parabola to the LEFT when it has a plus sign?

This confuses many students! Think of it this way: (x + 3)² = 0 when x = -3. The parabola reaches its minimum (vertex) where the expression inside equals zero. So x + 3 = 0 means x = -3, placing the vertex at x = -3.

How do I remember which direction horizontal shifts go?

Use the "opposite rule": The sign inside the parentheses is opposite to the shift direction. (x + 3) shifts LEFT 3 units, (x - 3) shifts RIGHT 3 units. Always solve for when the expression equals zero!

What does 'positive in all areas except x = -3' really mean?

This means the parabola touches the x-axis at exactly one point: x = -3. Everywhere else, the y-values are above the x-axis (positive). The vertex must be at (-3, 0) with the parabola opening upward.

Could the answer be y = -(x + 3)² since it's also zero at x = -3?

No! While y = -(x + 3)² is zero at x = -3, it's negative everywhere else (opens downward). We need the parabola to be positive except at x = -3, so it must open upward.

How can I quickly check my answer?

Test three points: x = -3 (should give y = 0), x = -2 (should be positive), and x = -4 (should be positive). For y = (x + 3)²: at x = -2, y = 1 ✓; at x = -4, y = 1 ✓

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