Standard 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.

We will begin by using the distributive property in order to expand the following expression.

(a+1)⋆(b+2) = ab+2a+b+2

We will then proceed to insert the known values into the equation and solve as follows:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

To find the standard representation of the quadratic function $f(x) = (x - 6)(x - 2)$, follow these steps:

• Step 1: Apply the FOIL method to expand the expression.
  - First: Multiply the first terms: $x \times x = x^2$.
  - Outside: Multiply the outer terms: $x \times (-2) = -2x$.
  - Inside: Multiply the inner terms: $(-6) \times x = -6x$.
  - Last: Multiply the last terms: $(-6) \times (-2) = 12$.
• Step 2: Combine the results from the FOIL method.
  - Combine all the expanded terms: $x^2 - 2x - 6x + 12$.
• Step 3: Simplify by combining like terms.
  - Combine the $x$-terms: $-2x - 6x = -8x$.
  - The expanded and simplified form is: $f(x) = x^2 - 8x + 12$.

By expanding and simplifying the given product, we have converted it to its standard form. Therefore, the standard representation of the function is $f(x) = x^2 - 8x + 12$.

The correct choice from the provided options is choice 2: $f(x) = x^2 - 8x + 12$.

To find the standard representation of the quadratic function given by $f(x) = (x+2)(x-4)$, we will expand the expression using the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last):

• First: Multiply the first terms in each binomial: $x \cdot x = x^2$.
• Outer: Multiply the outer terms in the binomials: $x \cdot (-4) = -4x$.
• Inner: Multiply the inner terms: $2 \cdot x = 2x$.
• Last: Multiply the last terms in each binomial: $2 \cdot (-4) = -8$.

Now, let's combine these results:

The expression becomes $x^2 - 4x + 2x - 8$.

Next, we combine like terms:

The terms involving $x$ are $-4x + 2x$, which simplifies to $-2x$.

Thus, the expression simplifies to: $f(x) = x^2 - 2x - 8$

Upon comparing this result to the provided choices, we find that it matches choice 3.

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

To find the standard representation of the quadratic function $f(x) = 3x(x + 4)$, follow these steps:

• Step 1: Apply the distributive property to expand the expression $x(x + 4)$.
  Using this property, we have:
  $x(x + 4) = x \cdot x + x \cdot 4 = x^2 + 4x$.
• Step 2: Multiply each term by the coefficient outside the parenthesis, which is 3.
  This gives us:
  $3(x^2 + 4x) = 3 \cdot x^2 + 3 \cdot 4x$.
• Step 3: Simplify by performing the multiplication.
  $3x^2 + 12x$.

Therefore, the standard representation of the function is $f(x) = 3x^2 + 12x$. This matches choice 3 in the provided answers.

To solve this problem, we'll convert the given function from its factored form to the standard form using the distributive property. The given function is $f(x) = -x(x - 8)$.

Let's go through the necessary steps:

• Step 1: Apply the distributive property to expand the expression.
  $f(x) = -x(x - 8) = -x \cdot x + (-x) \cdot (-8)$
• Step 2: Simplify each term.
  $-x \cdot x = -x^2$ and $(-x) \cdot (-8) = +8x$.
• Step 3: Combine the terms to express $f(x)$ in standard form:
  $f(x) = -x^2 + 8x$.

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

Comparing this result to the multiple-choice options, we can see that the correct choice is option 3: $f(x)=-x^2+8x$.

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.

To solve this problem and find the standard representation of the function $f(x) = (x+1)(x-1)$, we will expand the product using the distributive property, often recalled as FOIL (First, Outer, Inner, Last) for the product of two binomials.

Let's proceed step-by-step:

• Step 1: Apply the distributive property:
  $f(x) = (x+1)(x-1)$ would become:
• First terms: $x \cdot x = x^2$
• Outer terms: $x \cdot (-1) = -x$
• Inner terms: $1 \cdot x = x$
• Last terms: $1 \cdot (-1) = -1$

Step 2: Combine all the terms obtained from the FOIL method:
$x^2 - x + x - 1$

Step 3: Simplify the expression by combining like terms:
The terms $-x$ and $x$ cancel each other out, simplifying to:
$f(x) = x^2 - 1$

Thus, the standard representation of the function is $f(x) = x^2 - 1$.

To find the standard representation of the function $f(x) = (2x + 1)(x - 2)$, we'll follow these steps to expand and simplify the expression:

• Step 1: Distribute each term of the first binomial over each term of the second binomial using the FOIL method.
• Step 2: Combine like terms to express the function in standard quadratic form.

Now, let's expand the expression:
1. Multiply the first terms: $2x \cdot x = 2x^2$
2. Multiply the outer terms: $2x \cdot (-2) = -4x$
3. Multiply the inner terms: $1 \cdot x = x$
4. Multiply the last terms: $1 \cdot (-2) = -2$

Next, we combine these results:
- The $2x^2$ term remains as is.
- Add the linear terms: $-4x + x = -3x$
- The constant term is $-2$.

Thus, the expanded and simplified form of the function is:
$f(x) = 2x^2 - 3x - 2$

The final expression in standard form is $f(x) = 2x^2 - 3x - 2$.

To find the standard form of the given quadratic function $f(x) = (x+3)(-x-4)$, we will expand it using the distributive property.

Step 1: Expand the product.
Using the distributive property (or FOIL method):

$f(x) = (x+3)(-x-4)$

Apply distribution:
First: $x \cdot -x = -x^2$
Outside: $x \cdot -4 = -4x$
Inside: $3 \cdot -x = -3x$
Last: $3 \cdot -4 = -12$

Step 2: Combine all terms together:

$f(x) = -x^2 - 4x - 3x - 12$

Step 3: Simplify by combining like terms:
Combine the $x$ terms:

$f(x) = -x^2 - 7x - 12$

Therefore, the standard representation of the function is $f(x) = -x^2 - 7x - 12$.

The correct choice from the given options is choice 4.

$f(x)=-x^2-7x-12$

To solve the problem of finding the product (factored) representation of the quadratic function $f(x) = x^2 - 7x + 12$, we proceed as follows:

• Step 1: Identify the function, which is $f(x) = x^2 - 7x + 12$.
• Step 2: We need to factor this quadratic expression. We're looking for two numbers whose product is 12 and whose sum is -7.
• Step 3: The factor pairs of 12 are $(1, 12)$, $(2, 6)$, $(3, 4)$, including negative pairs because the sum must be negative.
• Step 4: Consider the pair $(-3, -4)$. The product $(-3) \times (-4)$ equals 12, and the sum $(-3) + (-4)$ equals -7.

Therefore, the factors of the quadratic expression are $x - 3$ and $x - 4$. This implies that the function $f(x)$ can be expressed in product form as:

$f(x) = (x - 3)(x - 4)$

This means the correct factorization is $(x - 3)(x - 4)$, which corresponds to choice 3 from the given options.

Thus, the representation of the product of the function is $(x - 3)(x - 4)$.

The problem requires finding the product representation of the quadratic function $f(x) = x^2 - 2x - 3$.

Let's execute the factorization of the quadratic equation:

• The standard form for the function is $f(x) = ax^2 + bx + c$. Here, $a = 1$, $b = -2$, $c = -3$.
• We seek two numbers that multiply to $c = -3$ and sum to $b = -2$.
• Checking possible integer pairs: $(-3, 1)$ can accomplish this, since $-3 \times 1 = -3$ and $-3 + 1 = -2$.
• The factorization becomes $f(x) = (x - 3)(x + 1)$.

To verify, we can expand the binomials:

$(x - 3)(x + 1) = x^2 + x - 3x - 3 = x^2 - 2x - 3$.

This matches the original polynomial, confirming the product representation is correct.

In conclusion, the factorization or product representation of the given quadratic function is $\mathbf{(x-3)(x+1)}$.

To solve the problem of factoring the quadratic expression $f(x) = x^2 - 3x - 18$, we will use the following method:

• Step 1: Identify and understand the quadratic expression, which is given in standard form: $ax^2 + bx + c$. For this expression, $a = 1$, $b = -3$, and $c = -18$.
• Step 2: Compute the product of $a$ and $c$, which yields $1 \cdot (-18) = -18$. We need to find two numbers whose product is $-18$ and whose sum is $-3$.
• Step 3: Look for pairs of factors of $-18$: - $1, -18$ - $-1, 18$ - $2, -9$ - $-2, 9$ - $3, -6$ - $-3, 6$

Among these, the pair $(3, -6)$ adds up to $-3$ and multiplies to $-18$.

• Step 4: Rewrite the quadratic expression using these numbers to represent the middle term:
  $x^2 - 3x - 18 = x^2 + 3x - 6x - 18$.
• Step 5: Group the terms to facilitate factoring:
  $(x^2 + 3x) + (-6x - 18)$.
• Step 6: Factor out the common factors in each grouped terms:
  $x(x + 3) - 6(x + 3)$.
• Step 7: Factor out the common binomial:
  $(x - 6)(x + 3)$.

Therefore, the factorized form of the quadratic function $f(x) = x^2 - 3x - 18$ is $(x - 6)(x + 3)$.

To determine the product representation of $f(x) = x^2 + x - 2$, we can factor the quadratic equation by following these steps:

• Step 1: Identify the product $ac = 1 \times (-2) = -2$ and sum $b = 1$.
• Step 2: Find two numbers that multiply to $-2$ and add to $1$. These numbers are $2$ and $-1$.
• Step 3: Rewrite the middle term using these numbers: $x^2 + 2x - 1x - 2$.
• Step 4: Factor by grouping:
  - Group $x^2 + 2x$ and $-1x - 2$ as separate pairs:
  - $x(x + 2) - 1(x + 2)$.
• Step 5: Factor out the common terms:
  $(x + 2)(x - 1)$.

Thus, the product representation of the function is $(x + 2)(x - 1)$.

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