Vertex Representation: Transition between different representations of a function

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

• Expand $(x-3)^2$ using the formula for a square:
  $(x-3)^2 = x^2 - 2 \times x \times 3 + 3^2 = x^2 - 6x + 9$
• Add the $x$ from $f(x) = (x-3)^2 + x$ to the expanded terms:
  $x^2 - 6x + 9 + x$
• Combine like terms:
  $x^2 - 6x + x + 9 = x^2 - 5x + 9$

Therefore, the standard form of the function $f(x)$ is $f(x) = x^2 - 5x + 9$.

Thus, the correct choice is Choice 3.

To convert $f(x) = (2x+1)^2 - 1$ into its standard quadratic form, we need to expand $(2x+1)^2$ first and then adjust for the subtraction of 1.

The expansion is carried out using the binomial expansion formula:

$(2x + 1)^2 = (2x)^2 + 2(2x)(1) + 1^2$.

Calculating each term gives:

• $(2x)^2 = 4x^2$
• $2(2x)(1) = 4x$
• $1^2 = 1$

Combining these, we obtain:

$(2x + 1)^2 = 4x^2 + 4x + 1$

Now, substituting back into the original equation:

$f(x) = (2x+1)^2 - 1 = (4x^2 + 4x + 1) - 1$

Subtracting 1 from the constant term, we get:

$f(x) = 4x^2 + 4x + 1 - 1 = 4x^2 + 4x$

Therefore, the standard form representation of the function is $f(x) = 4x^2 + 4x$.

To convert the function $f(x) = (-x + 1)^2 + 3$ to its standard form, follow these steps:

Step 1: Expand the binomial $(-x + 1)^2$.
$(-x + 1)^2 = (-x)^2 + 2(-x)(1) + 1^2$

This simplifies to:
$(-x)^2 = x^2$
$2(-x)(1) = -2x$
$1^2 = 1$

Combining these terms gives:
$(-x + 1)^2 = x^2 - 2x + 1$

Step 2: Add the constant term $+3$ to the expanded form:
$f(x) = (x^2 - 2x + 1) + 3$

Step 3: Simplify the expression:
$f(x) = x^2 - 2x + 1 + 3 = x^2 - 2x + 4$

Thus, the standard representation of the function is $f(x) = x^2 - 2x + 4$.

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

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

To convert the quadratic function from vertex form to standard form, execute the following steps:

• Step 1: Begin with the given vertex form $f(x) = (x-5)^2 - 10$.
• Step 2: Expand $(x-5)^2$ using the formula $(a-b)^2 = a^2 - 2ab + b^2$, which results in:

$(x-5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2 = x^2 - 10x + 25$.

• Step 3: Replace the expanded form into the original function:

$f(x) = x^2 - 10x + 25 - 10$.

• Step 4: Combine like terms:

$f(x) = x^2 - 10x + 15$.

Therefore, the standard form of the function is $f(x) = x^2 - 10x + 15$.

Comparing with the given choices, the correct option is:

Choice 2: $f(x) = x^2 - 10x + 15$

To convert the given quadratic function into its standard form, follow these steps:

• Step 1: Expand the Binomial
  We begin with the function in vertex form: $f(x) = (x + 5)^2 + 3$. The expression $(x + 5)^2$ can be expanded using the binomial theorem: $(a + b)^2 = a^2 + 2ab + b^2$.

• Step 2: Apply the Expansion Formula
  Let $a = x$ and $b = 5$. Therefore, $(x + 5)^2 = x^2 + 2 \times x \times 5 + 5^2 = x^2 + 10x + 25$.

• Step 3: Add the Constant
  Now, add the constant 3 to this expanded result: $x^2 + 10x + 25 + 3 = x^2 + 10x + 28$.

Thus, the standard representation of the function is $f(x) = x^2 + 10x + 28$.

Given the choices, the correct answer is $f(x) = x^2 + 10x + 28$, which matches choice 2.

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

To find the standard representation of the quadratic function $f(x)=(-x-2)^2-5$, we'll proceed with the following steps:

• Step 1: Expand the quadratic expression $(-x-2)^2$.
• Step 2: Simplify the expression resulting from the expansion.
• Step 3: Subtract the constant term $-5$.

Let's execute these steps in detail:

Step 1: Expand the expression $(-x-2)^2$.
To expand, use the formula $(a+b)^2 = a^2 + 2ab + b^2$, where $a=-x$ and $b=-2$.
$(-x-2)^2 = (-x)^2 + 2(-x)(-2) + (-2)^2$.

Step 2: Calculate the expanded form.
$(-x)^2 = x^2$,
$2(-x)(-2) = 4x$,
and $(-2)^2 = 4$.
Combining these, we have:
$x^2 + 4x + 4$.

Step 3: Incorporate the constant from the original function.
The original function is $f(x)=(-x-2)^2-5$. Thus, we subtract 5 from the expanded result:
$f(x) = x^2 + 4x + 4 - 5$, which simplifies to:
$f(x) = x^2 + 4x - 1$.

Therefore, the standard form of the given quadratic function is $f(x) = x^2 + 4x -1$.

To solve this problem, we'll transform the given expression into standard quadratic form by expanding and simplifying:

• Step 1: Expand $(x - 6)^2$
  $(x - 6)^2 = x^2 - 12x + 36$
• Step 2: Add $2x$
  $f(x) = x^2 - 12x + 36 + 2x$
• Step 3: Simplify
  Combine like terms: $-12x + 2x = -10x$, resulting in the expression:
  $f(x) = x^2 - 10x + 36$

Therefore, the standard form of the function $f(x)$ is $x^2 - 10x + 36$. This corresponds to choice 1 in the given list.

Thus, the final solution is $f(x) = x^2 - 10x + 36$.