Convert Vertex Form to Standard Form: Explore (x-2)² + 4

Q: Why can't I just distribute the square to get $ x^2 - 2^2 $ ?

That's not how squaring works! $ (x-2)^2 $ means $ (x-2) \times (x-2) $, not $ x^2 - 2^2 $. You must use the formula $ (a-b)^2 = a^2 - 2ab + b^2 $ or multiply it out fully.

Q: How do I remember the perfect square formula?

Think FOIL backwards! For $ (x-2)^2 $: First terms give $ x^2 $, Outer + Inner give $ -2x - 2x = -4x $, Last terms give $ +4 $. The middle term is always twice the product of the two terms.

Q: What's the difference between vertex form and standard form?

Vertex form like $ f(x) = (x-2)^2 + 4 $ shows the vertex clearly at (2,4). Standard form like $ f(x) = x^2 - 4x + 8 $ makes it easy to see the coefficients and y-intercept.

Q: Why do I get 8 instead of 4 for the constant term?

You're adding two separate +4 terms! One comes from expanding $ (x-2)^2 = x^2 - 4x + 4 $, and the other is the +4 from the original function. So: 4 + 4 = 8.

Q: Can I check my answer by plugging in a value?

Absolutely! Try x = 0: Original gives $ (0-2)^2 + 4 = 4 + 4 = 8 $. Your standard form should give $ 0^2 - 4(0) + 8 = 8 $. If they match, you're correct!

Quadratic Expansion with Perfect Square Binomials

Find the standard representation of the following function

$f(x)=(x-2)^2+4$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify to the standard representation of the function

00:04 Expand brackets according to short multiplication formulas

00:10 Calculate powers and multiplications

00:24 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 standard representation of the following function

$f(x)=(x-2)^2+4$

Step-by-step solution

We need to convert the given function $f(x) = (x-2)^2 + 4$ to standard form.

To expand $(x-2)^2$ , we use the formula $(a-b)^2 = a^2 - 2ab + b^2$ . Applying this to $(x-2)^2$ , we get:

$(x-2)^2 = x^2 - 4x + 4$ .

This accounts for the expanded square. Next, we add the constant term $4$ from the original function $(x-2)^2 + 4$ :

$f(x) = x^2 - 4x + 4 + 4$ .

Simplify by combining the constant terms:

$f(x) = x^2 - 4x + 8$ .

The standard form of the function is thus $f(x) = x^2 - 4x + 8$ .

Final Answer

$f(x)=x^2-4x+8$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Use $(a-b)^2 = a^2 - 2ab + b^2$ for perfect squares
Technique: $(x-2)^2 = x^2 - 4x + 4$ then add constant
Check: Verify by substituting x = 0: both forms give f(0) = 8 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to add the constant term after expansion
Don't expand $(x-2)^2$ and stop at $x^2 - 4x + 4$ = missing the +4! This ignores the constant term from the original function and gives $f(x) = x^2 - 4x + 4$ instead of the correct $f(x) = x^2 - 4x + 8$ . Always add all terms from the original function after expanding.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=2,b=2,c=2 $$

FAQ

Everything you need to know about this question

Why can't I just distribute the square to get $x^2 - 2^2$ ?

That's not how squaring works! $(x-2)^2$ means $(x-2) \times (x-2)$ , not $x^2 - 2^2$ . You must use the formula $(a-b)^2 = a^2 - 2ab + b^2$ or multiply it out fully.

How do I remember the perfect square formula?

Think FOIL backwards! For $(x-2)^2$ : First terms give $x^2$ , Outer + Inner give $-2x - 2x = -4x$ , Last terms give $+4$ . The middle term is always twice the product of the two terms.

What's the difference between vertex form and standard form?

Vertex form like $f(x) = (x-2)^2 + 4$ shows the vertex clearly at (2,4). Standard form like $f(x) = x^2 - 4x + 8$ makes it easy to see the coefficients and y-intercept.

Why do I get 8 instead of 4 for the constant term?

You're adding two separate +4 terms! One comes from expanding $(x-2)^2 = x^2 - 4x + 4$ , and the other is the +4 from the original function. So: 4 + 4 = 8.

Can I check my answer by plugging in a value?

Absolutely! Try x = 0: Original gives $(0-2)^2 + 4 = 4 + 4 = 8$ . Your standard form should give $0^2 - 4(0) + 8 = 8$ . If they match, you're correct!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Ways of Representing the Quadratic Function questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Convert Vertex Form to Standard Form: Explore (x-2)² + 4

Quadratic Expansion with Perfect Square Binomials

❤️ 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 can't I just distribute the square to get $x^2 - 2^2$ ?

How do I remember the perfect square formula?

What's the difference between vertex form and standard form?

Why do I get 8 instead of 4 for the constant term?

Can I check my answer by plugging in a value?

🌟 Unlock Your Math Potential

More Questions

Vertex Representation

Transforming Quadratics: Find the Standard Form of f(x) = (x-3)² + x

Convert the Function f(x)=(x-2)²+3 to its Standard Form

Find the Standard Form of the Function Equation: Rewrite f(x) = (-x+1)²+3

Find the Standard Representation of (2x+1)² - 1

Standard Representation

Convert the Quadratic Expression (-x-2)² - 5 to Its Standard Form

Find the Standard Form of the Function: Transforming (x+4)² - 16

Convert Vertex Form to Standard Form: Explore (x-2)² + 4

Quadratic Expansion with Perfect Square Binomials

❤️ 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 can't I just distribute the square to get x2−22 x^2 - 2^2 x2−22?

How do I remember the perfect square formula?

What's the difference between vertex form and standard form?

Why do I get 8 instead of 4 for the constant term?

Can I check my answer by plugging in a value?

🌟 Unlock Your Math Potential

More Questions

Vertex Representation

Transforming Quadratics: Find the Standard Form of f(x) = (x-3)² + x

Convert the Function f(x)=(x-2)²+3 to its Standard Form

Find the Standard Form of the Function Equation: Rewrite f(x) = (-x+1)²+3

Find the Standard Representation of (2x+1)² - 1

Standard Representation

Convert the Quadratic Expression (-x-2)² - 5 to Its Standard Form

Find the Standard Form of the Function: Transforming (x+4)² - 16

Why can't I just distribute the square to get $x^2 - 2^2$ ?