Calculate the Vertex of y = (x-3)² + 6

Q: Why is the vertex (3,6) and not (-3,6)?

Great question! In vertex form $ y = (x-h)^2 + k $, the h-value is the opposite of what appears after x. Since we have $ (x-3)^2 $, h = 3, not -3!

Q: What if the equation was y = (x+3)² + 6 instead?

Then the vertex would be (-3,6)! Remember: $ (x+3)^2 $ is the same as $ (x-(-3))^2 $, so h = -3.

Q: Can I solve this by expanding the equation first?

You could expand to standard form and use $ x = -\frac{b}{2a} $, but that's much more work! When given vertex form, read the vertex directly - it's much faster.

Q: How do I know this parabola opens upward?

Since there's no negative sign in front of $ (x-3)^2 $, the coefficient of the squared term is positive (+1), so the parabola opens upward with (3,6) as its minimum point.

Q: What does the vertex tell me about the graph?

The vertex (3,6) is the turning point of the parabola. It's the lowest point since this parabola opens upward, and x = 3 is the axis of symmetry where the graph can be folded in half.

Vertex Form with Direct Identification

Find the vertex of the parabola

$y=(x-3)^2+6$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the vertex of the parabola

00:03 Use the formula to describe the parabola function

00:11 The coordinates of the vertex are (P,K)

00:20 Use this formula and find the vertex point

00:24 Substitute appropriate values according to the given data

00:27 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 vertex of the parabola

$y=(x-3)^2+6$

Step-by-step solution

To solve the problem of finding the vertex of the parabola $y = (x-3)^2 + 6$ , we take the following steps:

Step 1: Identify the form of the given equation.
The equation is given in the vertex form of a quadratic function, which is generally expressed as $y = (x-h)^2 + k$ .

Step 2: Recognize the coefficients.
In the given equation $y = (x-3)^2 + 6$ , compare it with the standard form $y=(x-h)^2 + k$ to identify $h$ and $k$ . Here, $h = 3$ and $k = 6$ .

Step 3: Determine the vertex.
The vertex of the parabola, therefore, is directly given by the point $(h, k) = (3, 6)$ .

As a conclusion, the vertex of the parabola described by the equation $y = (x-3)^2 + 6$ is located at the point $(3, 6)$ .

Final Answer

$(3,6)$

Key Points to Remember

Essential concepts to master this topic

Standard Form: Vertex form is $y = (x-h)^2 + k$ with vertex at (h,k)
Pattern Recognition: From $y = (x-3)^2 + 6$ , identify h = 3 and k = 6
Verification: Check that h = 3 is axis of symmetry and k = 6 is minimum value ✓

Common Mistakes

Avoid these frequent errors

Sign confusion when reading h-value from vertex form
Don't read $(x-3)^2$ as h = -3! This gives vertex (-3,6) instead of (3,6). The negative sign is built into the form, so h equals the opposite of what's inside. Always remember: $(x-h)^2$ means h is the opposite of the number after x.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

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

Calculate point C.

FAQ

Everything you need to know about this question

Why is the vertex (3,6) and not (-3,6)?

Great question! In vertex form $y = (x-h)^2 + k$ , the h-value is the opposite of what appears after x. Since we have $(x-3)^2$ , h = 3, not -3!

How do I remember which number is x and which is y in the vertex?

Easy trick: The number with x goes with x, and the number by itself goes with y. So from $(x-3)^2 + 6$ , the vertex is (3,6) - first number with x-coordinate, second with y-coordinate.

What if the equation was y = (x+3)² + 6 instead?

Then the vertex would be (-3,6)! Remember: $(x+3)^2$ is the same as $(x-(-3))^2$ , so h = -3.

Can I solve this by expanding the equation first?

You could expand to standard form and use $x = -\frac{b}{2a}$ , but that's much more work! When given vertex form, read the vertex directly - it's much faster.

How do I know this parabola opens upward?

Since there's no negative sign in front of $(x-3)^2$ , the coefficient of the squared term is positive (+1), so the parabola opens upward with (3,6) as its minimum point.

What does the vertex tell me about the graph?

The vertex (3,6) is the turning point of the parabola. It's the lowest point since this parabola opens upward, and x = 3 is the axis of symmetry where the graph can be folded in half.

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Calculate the Vertex of y = (x-3)² + 6

Vertex Form with Direct Identification

❤️ 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 the vertex (3,6) and not (-3,6)?

How do I remember which number is x and which is y in the vertex?

What if the equation was y = (x+3)² + 6 instead?

Can I solve this by expanding the equation first?

How do I know this parabola opens upward?

What does the vertex tell me about the graph?

🌟 Unlock Your Math Potential

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