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

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

Vertex form is f(x) = a(x-h)² + k and shows the vertex clearly at (h,k). Standard form is f(x) = ax² + bx + c and makes it easier to identify the y-intercept and use the quadratic formula.

Q: Why do I need to expand (x-2)²?

Expanding lets you see all the terms clearly! $ (x-2)^2 $ hides the middle term -4x that becomes visible when you expand to $ x^2 - 4x + 4 $.

Q: How do I remember the (a-b)² formula?

Think "First squared, minus twice the product, plus second squared": $ (x-2)^2 = x^2 - 2(x)(2) + 2^2 = x^2 - 4x + 4 $

Q: What if I get the wrong constant term?

Double-check your arithmetic! From $ (x-2)^2 + 3 $, you get $ x^2 - 4x + 4 + 3 $. Make sure to add: 4 + 3 = 7.

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

Absolutely! Try x = 0: Original gives $ (0-2)^2 + 3 = 4 + 3 = 7 $. Your standard form should give $ 0^2 - 4(0) + 7 = 7 $ ✓

Quadratic Functions with Vertex to Standard Form

Find the standard representation of the following function

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplified to the standard representation of the function

00:03 Open parentheses according to the shortened multiplication formulas

00:12 Calculate powers and products

00:19 Add

00:25 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

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

Step-by-step solution

To convert the function from vertex form to standard form, follow these steps:

Step 1: Identify the vertex form - $f(x) = (x-2)^2 + 3$ . The terms inside the parentheses represent a perfect square trinomial.
Step 2: Expand the square. Recall: $(a-b)^2 = a^2 - 2ab + b^2$ . Here, $a = x$ and $b = 2$ .
Step 3: Expand $(x-2)^2$ :
$(x-2)^2 = x^2 - 2 \cdot x \cdot 2 + 2^2 = x^2 - 4x + 4$ .
Step 4: Add the constant from the original function:
$f(x) = (x^2 - 4x + 4) + 3 = x^2 - 4x + 7$ .

After expanding and simplifying, we find that $f(x) = x^2 - 4x + 7$ is the standard form of the function.

Therefore, the correct choice that matches this solution is choice 3, which is $f(x) = x^2 - 4x + 7$ .

Final Answer

$f(x)=x^2-4x+7$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Use (a-b)² = a² - 2ab + b² formula
Technique: (x-2)² = x² - 4x + 4, then add constant term
Check: Standard form ax² + bx + c should equal original when factored ✓

Common Mistakes

Avoid these frequent errors

Forgetting to add the constant term after expanding
Don't expand (x-2)² = x² - 4x + 4 and stop there = missing the +3! This gives x² - 4x + 4 instead of x² - 4x + 7. Always add all constant terms outside the parentheses after expanding.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=3,b=0,c=-3 $$

FAQ

Everything you need to know about this question

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

Vertex form is f(x) = a(x-h)² + k and shows the vertex clearly at (h,k). Standard form is f(x) = ax² + bx + c and makes it easier to identify the y-intercept and use the quadratic formula.

Why do I need to expand (x-2)²?

Expanding lets you see all the terms clearly! $(x-2)^2$ hides the middle term -4x that becomes visible when you expand to $x^2 - 4x + 4$ .

How do I remember the (a-b)² formula?

Think "First squared, minus twice the product, plus second squared": $(x-2)^2 = x^2 - 2(x)(2) + 2^2 = x^2 - 4x + 4$

What if I get the wrong constant term?

Double-check your arithmetic! From $(x-2)^2 + 3$ , you get $x^2 - 4x + 4 + 3$ . Make sure to add: 4 + 3 = 7.

Can I check my answer by plugging in a value?

Absolutely! Try x = 0: Original gives $(0-2)^2 + 3 = 4 + 3 = 7$ . Your standard form should give $0^2 - 4(0) + 7 = 7$ ✓

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

Quadratic Functions with Vertex to Standard Form

❤️ 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

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

Why do I need to expand (x-2)²?

How do I remember the (a-b)² formula?

What if I get the wrong constant term?

Can I check my answer by plugging in a value?

🌟 Unlock Your Math Potential

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