Finding the Axis of Symmetry: y=(x-5)²+15 in Vertex Form

Q: Why is the axis x = 5 and not x = -5?

In vertex form $ y = (x - h)^2 + k $, the axis is x = h. Since we have $ (x - 5)^2 $, here h = 5, so the axis is x = 5. The negative sign is already built into the form!

Q: Do I need to expand the equation to find the axis?

No! That's the beauty of vertex form - you can read the axis directly. The explanation shows expansion, but it's unnecessary. From $ y = (x - 5)^2 + 15 $, the axis is immediately x = 5.

Q: What does the +15 tell me?

The +15 is the k-value in vertex form, representing the y-coordinate of the vertex. So the complete vertex is (5, 15), but the axis of symmetry only uses the x-coordinate: x = 5.

Q: How can I remember the vertex form pattern?

Think of it as $ y = (x - \text{axis})^2 + \text{height} $. Whatever number comes after the minus sign in the parentheses is your axis of symmetry!

Q: Would this work for any vertex form equation?

Yes! Whether it's $ (x - 3)^2 $, $ (x + 7)^2 $, or $ (x - 0)^2 $, just identify the h-value. Remember: $ (x + 7)^2 $ means $ (x - (-7))^2 $, so h = -7.

Vertex Form with Axis Identification

What is the axis of symmetry of the equation?

$y=(x-5)^2+15$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the axis of symmetry of the function

00:05 We'll use the formula for representing a parabola

00:15 The axis of symmetry is on the X value of the vertex point (P)

00:19 We'll use the formula and find the term P

00:25 This is the axis of symmetry

00:28 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

What is the axis of symmetry of the equation?

$y=(x-5)^2+15$

Step-by-step solution

The first step in solving the equation you presented:

y=(x-5)²+15

is to expand the parentheses:

y=x²-10x+25+15

y=x²-10x+40

From here, we can use the formula to find the X-coordinate of the vertex:

-b/2a

Let's substitute the values from the equation:

-(-10)/2*1 =

10/2=5

The axis of symmetry of the parabola is X=5

Final Answer

$x=5$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: For y = (x - h)² + k, axis is x = h
Direct Reading: From (x - 5)² + 15, h = 5 so axis is x = 5
Verification: Substitute x = 5: (5-5)² + 15 = 15, confirming vertex location ✓

Common Mistakes

Avoid these frequent errors

Confusing the sign in vertex form
Don't think (x - 5)² means axis is x = -5! The minus sign is already in the form, so h = 5 directly. Always remember: (x - h)² means the axis is x = h, not x = -h.

Practice Quiz

Test your knowledge with interactive questions

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$$ f(x)=-3x^2+3 $$

FAQ

Everything you need to know about this question

Why is the axis x = 5 and not x = -5?

In vertex form $y = (x - h)^2 + k$ , the axis is x = h. Since we have $(x - 5)^2$ , here h = 5, so the axis is x = 5. The negative sign is already built into the form!

Do I need to expand the equation to find the axis?

No! That's the beauty of vertex form - you can read the axis directly. The explanation shows expansion, but it's unnecessary. From $y = (x - 5)^2 + 15$ , the axis is immediately x = 5.

What does the +15 tell me?

The +15 is the k-value in vertex form, representing the y-coordinate of the vertex. So the complete vertex is (5, 15), but the axis of symmetry only uses the x-coordinate: x = 5.

How can I remember the vertex form pattern?

Think of it as $y = (x - \text{axis})^2 + \text{height}$ . Whatever number comes after the minus sign in the parentheses is your axis of symmetry!

Would this work for any vertex form equation?

Yes! Whether it's $(x - 3)^2$ , $(x + 7)^2$ , or $(x - 0)^2$ , just identify the h-value. Remember: $(x + 7)^2$ means $(x - (-7))^2$ , so h = -7.

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