Vertex form of the quadratic equation

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 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 solve for the vertex of the parabola given by the equation $y = (x-3)^2 - 1$, we start by comparing the equation with the standard vertex form of a quadratic function: $y = a(x-h)^2 + k$.

In the given equation, $y = (x-3)^2 - 1$, we identify:
- $h = 3$, which corresponds to the horizontal shift of the parabola.
- $k = -1$, which represents the vertical shift.

Therefore, the vertex of the parabola is at the point $(h, k)$, which is $(3, -1)$.

Thus, the vertex of the parabola is $(3, -1)$.

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