Determine the Vertex of the Parabola: y = (x-3)² - 1

Vertex Form with Direct Identification

Find the vertex of the parabola

y=(x3)21 y=(x-3)^2-1

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Step-by-step video solution

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00:00 Find the vertex of the parabola
00:03 Use the formula to describe the parabola function
00:07 The coordinates of the vertex are (P,K)
00:11 Use this formula and find the vertex point
00:14 Substitute appropriate values according to the given data
00:19 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

Find the vertex of the parabola

y=(x3)21 y=(x-3)^2-1

2

Step-by-step solution

To solve for the vertex of the parabola given by the equation y=(x3)21 y = (x-3)^2 - 1 , we start by comparing the equation with the standard vertex form of a quadratic function: y=a(xh)2+k y = a(x-h)^2 + k .

In the given equation, y=(x3)21 y = (x-3)^2 - 1 , we identify:
- h=3 h = 3 , which corresponds to the horizontal shift of the parabola.
- k=1 k = -1 , which represents the vertical shift.

Therefore, the vertex of the parabola is at the point (h,k)(h, k), which is (3,1)(3, -1).

Thus, the vertex of the parabola is (3,1)(3, -1).

3

Final Answer

(3,1) (3,-1)

Key Points to Remember

Essential concepts to master this topic
  • Vertex Form: Standard form is y=a(xh)2+k y = a(x-h)^2 + k where vertex is (h,k)
  • Identification: From y=(x3)21 y = (x-3)^2 - 1 , h = 3 and k = -1
  • Check: Substitute x = 3: y=(33)21=01=1 y = (3-3)^2 - 1 = 0 - 1 = -1

Common Mistakes

Avoid these frequent errors
  • Confusing signs when identifying h and k values
    Don't think h = -3 from (x-3) = wrong vertex (-3,-1)! The negative sign is part of the pattern (x-h), so when you see (x-3), h is positive 3. Always remember: (x-h) means h is the opposite of the number you see.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

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

Calculate point C.

CCCAAABBB

FAQ

Everything you need to know about this question

Why is the x-coordinate 3 and not -3?

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Great question! In vertex form y=a(xh)2+k y = a(x-h)^2 + k , the pattern is (x - h). When you see (x - 3), it means h = 3, not -3. Think of it as "x minus 3 equals zero when x = 3".

What if the equation was y = (x+3)² - 1 instead?

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Then you'd rewrite it as y=(x(3))21 y = (x-(-3))^2 - 1 , so h = -3 and the vertex would be (-3, -1). The plus sign means h is negative!

How do I remember which coordinate is which?

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Use the pattern (h, k) where h comes from the x-part and k comes from the y-part. In (xh)2+k (x-h)^2 + k , h affects x-direction and k affects y-direction.

Can I check my vertex by graphing?

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Absolutely! The vertex is the lowest point (if a > 0) or highest point (if a < 0) of the parabola. Since our coefficient of the squared term is 1 (positive), (3, -1) should be the minimum point.

What if there's no number added or subtracted outside the parentheses?

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If you see something like y=(x2)2 y = (x-2)^2 , then k = 0! The vertex would be (2, 0). Missing terms just mean their value is zero.

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