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

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

Vertex form like $ (x+4)^2 - 16 $ shows the vertex clearly, while standard form like $ x^2 + 8x $ shows coefficients for graphing and calculations.

Q: Why do I need to expand the perfect square?

Expanding helps you see the individual coefficients of $ x^2 $, $ x $, and constant terms. This makes it easier to add, subtract, or compare with other quadratic functions.

Q: How do I remember the perfect square formula?

Think "First, Outer, Inner, Last" or remember the pattern: $ (a+b)^2 = a^2 + 2ab + b^2 $. The middle term is always twice the product of the two terms!

Q: What if I get confused with the signs?

Work step by step! In $ (x+4)^2 - 16 $, first expand the square completely: $ x^2 + 8x + 16 $, then subtract 16 at the very end.

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

Absolutely! Pick any x-value like x = 0. Check that both $ (0+4)^2 - 16 = 0 $ and $ 0^2 + 8(0) = 0 $ give the same result.

Quadratic Functions with Vertex Form Expansion

Find the standard representation of the following function

$f(x)=(x+4)^2-16$

❤️ 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 Simplified to the standard representation of the function

00:03 Open parentheses according to the shortened multiplication formulas

00:11 Calculate powers and products

00:24 Subtract

00:27 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

$f(x)=(x+4)^2-16$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Expand $(x + 4)^2$ using the formula $(a + b)^2 = a^2 + 2ab + b^2$ .
Step 2: Simplify the expression by subtracting 16 from the expanded result.
Step 3: Write the simplified expression in the standard form.

Now, let's work through each step:
Step 1: Start with the expression given in the problem:
$(x + 4)^2 = x^2 + 2 \cdot x \cdot 4 + 4^2$ .

This results in:
$x^2 + 8x + 16$ .

Step 2: Subtract 16 from the expanded expression:
$x^2 + 8x + 16 - 16 = x^2 + 8x$ .

Step 3: The standard form of the expression is now:
$f(x) = x^2 + 8x$ .

Therefore, the standard representation of the function is $f(x) = x^2 + 8x$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Perfect Square Formula: Use $(a+b)^2 = a^2 + 2ab + b^2$ to expand binomials
Technique: $(x+4)^2 = x^2 + 8x + 16$ then subtract constant
Check: Verify by factoring $x^2 + 8x = x(x+8)$ matches original vertex ✓

Common Mistakes

Avoid these frequent errors

Forgetting to subtract the constant term after expanding
Don't stop after expanding $(x+4)^2 = x^2 + 8x + 16$ and write that as your final answer = missing the -16! This gives $x^2 + 8x + 16$ instead of $x^2 + 8x$ . Always complete all operations in the original expression.

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

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

Vertex form like $(x+4)^2 - 16$ shows the vertex clearly, while standard form like $x^2 + 8x$ shows coefficients for graphing and calculations.

Why do I need to expand the perfect square?

Expanding helps you see the individual coefficients of $x^2$ , $x$ , and constant terms. This makes it easier to add, subtract, or compare with other quadratic functions.

How do I remember the perfect square formula?

Think "First, Outer, Inner, Last" or remember the pattern: $(a+b)^2 = a^2 + 2ab + b^2$ . The middle term is always twice the product of the two terms!

What if I get confused with the signs?

Work step by step! In $(x+4)^2 - 16$ , first expand the square completely: $x^2 + 8x + 16$ , then subtract 16 at the very end.

Can I check my answer by plugging in a value?

Absolutely! Pick any x-value like x = 0. Check that both $(0+4)^2 - 16 = 0$ and $0^2 + 8(0) = 0$ give the same result.

🌟 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

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

Quadratic Functions with Vertex Form Expansion

❤️ 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 the perfect square?

How do I remember the perfect square formula?

What if I get confused with the signs?

Can I check my answer by plugging in a value?

🌟 Unlock Your Math Potential

More Questions

Standard Representation

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

Transforming the Function: Reveal the Standard Form of f(x) = (x+5)² + 3

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

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

Vertex Representation

Convert (x-5)² - 10 to its Standard Quadratic Form

Express (x-6)² + 2x in Standard Function Form

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

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