Determine the Vertex for the Parabola in Equation: y = (x-7)² - 7

Q: Why is the vertex (7, -7) and not (-7, -7)?

Look at the pattern: $ y = a(x-h)^2 + k $. The h-value is what makes the parentheses equal zero. Since we have $ (x-7) $, we need x = 7 to get zero, so h = 7!

Q: What if the equation was y = (x+7)² - 7 instead?

Then you'd have $ y = (x-(-7))^2 + (-7) $, so h = -7 and k = -7. The vertex would be (-7, -7). Remember: (x+7) is the same as (x-(-7)).

Q: How do I know this is already in vertex form?

Vertex form has the pattern $ y = a(x-h)^2 + k $ with a perfect square and a constant. Since $ (x-7)^2 $ is already squared, no further work needed!

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

The vertex (7, -7) is the lowest point since the coefficient of the squared term is positive (+1). This parabola opens upward with its minimum at x = 7.

Q: Can I double-check my vertex by plugging in x = 7?

Absolutely! Substitute: $ y = (7-7)^2 - 7 = 0^2 - 7 = -7 $. So when x = 7, y = -7, confirming our vertex is (7, -7) ✓

Vertex Form with Coordinate Identification

Find the vertex of the parabola

$y=(x-7)^2-7$

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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 We'll use the formula to describe a parabolic function

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

00:14 We'll use this formula to find the vertex point

00:17 We'll substitute appropriate values according to the given data

00:20 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

$y=(x-7)^2-7$

Step-by-step solution

To solve this problem, we need to find the vertex of the given parabola $y = (x-7)^2 - 7$ . This equation is already in the vertex form of a parabola, $y = a(x-h)^2 + k$ . In this form, the vertex is given by the point $(h, k)$ .

Let's break it down:

The expression inside the parentheses, $(x-7)$ , shows that $h = 7$ .
The constant term outside, which is $-7$ , shows that $k = -7$ .

Therefore, the vertex of the parabola is given by the coordinates $(h, k)$ .
Substituting $h = 7$ and $k = -7$ into the vertex form, we have:

Vertex: $(7, -7)$

Therefore, the correct answer is choice 1: $(7, -7)$ .

Final Answer

$(7,-7)$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Pattern is y = a(x-h)² + k where vertex is (h,k)
Identification: From y = (x-7)² - 7, h = 7 and k = -7
Verification: Check that x-coordinate makes expression zero: (7-7)² = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing signs when identifying h-coordinate
Don't think (x-7) means h = -7! This gives wrong vertex (-7, -7) instead of (7, -7). The minus sign is already in the pattern, so h equals the number that makes the expression zero. Always identify h as the value that makes (x-h) = 0.

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 (7, -7) and not (-7, -7)?

Look at the pattern: $y = a(x-h)^2 + k$ . The h-value is what makes the parentheses equal zero. Since we have $(x-7)$ , we need x = 7 to get zero, so h = 7!

What if the equation was y = (x+7)² - 7 instead?

Then you'd have $y = (x-(-7))^2 + (-7)$ , so h = -7 and k = -7. The vertex would be (-7, -7). Remember: (x+7) is the same as (x-(-7)).

How do I know this is already in vertex form?

Vertex form has the pattern $y = a(x-h)^2 + k$ with a perfect square and a constant. Since $(x-7)^2$ is already squared, no further work needed!

What does the vertex tell me about the parabola?

The vertex (7, -7) is the lowest point since the coefficient of the squared term is positive (+1). This parabola opens upward with its minimum at x = 7.

Can I double-check my vertex by plugging in x = 7?

Absolutely! Substitute: $y = (7-7)^2 - 7 = 0^2 - 7 = -7$ . So when x = 7, y = -7, confirming our vertex is (7, -7) ✓

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Determine the Vertex for the Parabola in Equation: y = (x-7)² - 7

Vertex Form with Coordinate 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 (7, -7) and not (-7, -7)?

What if the equation was y = (x+7)² - 7 instead?

How do I know this is already in vertex form?

What does the vertex tell me about the parabola?

Can I double-check my vertex by plugging in x = 7?

🌟 Unlock Your Math Potential

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