Standard Representation - Examples, Exercises and Solutions

The goal is to express the quadratic equation $y = ax^2 + bx + c$ using the given parameters $a = -1$, $b = -1$, and $c = -1$.

First, substitute the values of $a$, $b$, and $c$ into the standard form:

• Substituting $a = -1$, the term becomes $-x^2$.
• Substituting $b = -1$, the term becomes $-x$.
• Substituting $c = -1$, the term remains $-1$.

Combine these terms to form the full expression:

$y = -x^2 - x - 1$

Therefore, the algebraic expression for the parameters $a = -1$, $b = -1$, and $c = -1$ is: $-x^2 - x - 1$.

Comparing with the given choices, the correct choice is option 4: $-x^2-x-1$

To solve the problem of creating an algebraic expression with the given parameters, we will proceed as follows:

• Step 1: Identify the given coefficients for the quadratic function, which are $a = 3$, $b = 0$, and $c = -3$.
• Step 2: Substitute these values into the standard quadratic expression $y = ax^2 + bx + c$.

Through substitution, the expression becomes:

$y = 3x^2 + 0x - 3$

We can further simplify this expression:

$y = 3x^2 - 3$

Thus, the algebraic expression with the given parameters is $y = 3x^2 - 3$.

The correct answer corresponds to choice number 1: $3x^2-3$.

Therefore, the solution to the problem is

$y = 3x^2 - 3$

To solve this problem, let's form the algebraic expression using the standard quadratic formula:

$y = ax^2 + bx + c$

Given are the values:
$a = 1$,
$b = -1$,
$c = 1$.

Substituting these values into the formula, we have:
$y = 1 \cdot x^2 + (-1) \cdot x + 1$

This simplifies to:
$y = x^2 - x + 1$

Thus, the algebraic expression is $\boldsymbol{x^2 - x + 1}$.

The correct choice from the given options is:

Choice 3: $x^2-x+1$

To solve this problem, we'll start by substituting the given parameters into the standard quadratic formula:

$y = ax^2 + bx + c$

The problem gives us the values:

• $a = -1$
• $b = -8$
• $c = 0$

This means we need to replace $a$, $b$, and $c$ in the formula:

$y = (-1)x^2 + (-8)x + 0$

Simplifying this expression further:

• The term with $a$: (-1)x^2\) results in $-x^2$.
• The term with $b$: (-8)x\) simplifies to $-8x$.
• The term with $c$: $0$ contributes nothing to the expression, so it is omitted.

Thus, the final algebraic expression is:

$y = -x^2 - 8x$

Therefore, the algebraic expression based on the given parameters is

$-x^2 - 8x$.

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

• Step 1: Substitute the given values $a = 3$, $b = 0$, and $c = 0$ into the quadratic function formula $y = ax^2 + bx + c$.
• Step 2: Simplify the expression.

Let's execute these steps:

Step 1: Substitute the values into the formula:
$y = 3x^2 + 0x + 0$

Step 2: Simplify the expression:
Eliminate the terms with zero coefficients to get:
$y = 3x^2$

Thus, the algebraic expression for the quadratic function with $a = 3$, $b = 0$, and $c = 0$ is $3x^2$.

Therefore, the correct choice from the options provided is choice 1: $3x^2$

To solve this problem, we need to form an algebraic expression for a quadratic function using given parameters.

We start by recalling the standard form of a quadratic function: $( ax^2 + bx + c )$. In this expression:

• $a$ is the coefficient of $x^2$
• $b$ is the coefficient of $x$
• $c$ is the constant term

Given the values are $a = 2$, $b = 4$, and $c = 8$, we substitute these into the standard form equation:

$ax^2 + bx + c = 2x^2 + 4x + 8$

This yields the algebraic expression for the quadratic function.

The correct expression, given all calculations and simplifications, is $2x^2 + 4x + 8$.

Referring to the choices provided, the correct choice is:

: $( 2x^2 + 4x + 8 )$

Therefore, the solution to the problem is $\boxed{2x^2 + 4x + 8}$.

To solve this problem, we'll follow the steps outlined:

• Step 1: Identify the given values for the quadratic function's parameters: $a = 2$, $b = \frac{1}{2}$, and $c = 4$.
• Step 2: Apply these values to the standard quadratic form $y = ax^2 + bx + c$.
• Step 3: Substitute the values to construct the algebraic expression.

Now, let's proceed with these steps:

Given the standard form of a quadratic expression $y = ax^2 + bx + c$:

Substituting the values, we obtain:

$y = 2x^2 + \frac{1}{2}x + 4$

Therefore, the correct algebraic expression for the quadratic function is $2x^2 + \frac{1}{2}x + 4$.

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

• Step 1: Formulate using the standard quadratic expression template.
• Step 2: Substitute the given parameters.
• Step 3: Simplify the resultant expression.

Now, let's work through each step:

Step 1: We use the standard form of a quadratic expression, which is $ax^2 + bx + c$.

Step 2: Substitute the values $a = 4$, $b = -16$, and $c = 0$ into this template:

$ax^2 + bx + c \rightarrow 4x^2 - 16x + 0$

Step 3: Simplify the expression:

The expression simplifies to $4x^2 - 16x$.

Thus, the algebraic expression based on the given parameters is $4x^2 - 16x$.

Checking against the answer choices, the correct choice is: $4x^2 - 16x$ .

To determine the algebraic expression, we start with the standard quadratic function:

$y = ax^2 + bx + c$

Given the values:

• $a = -1$
• $b = 1$
• $c = 0$

We substitute these into the formula:

$y = (-1)x^2 + 1x + 0$

Simplifying the expression gives:

$y = -x^2 + x$

Thus, the algebraic expression, when these parameters are substituted, is:

The solution to the problem is $\boxed{-x^2 + x}$.

To solve this problem, we'll use the following steps:

• Step 1: Substitute $a = 0$, $b = 1$, $c = 0$ into the quadratic equation $y = ax^2 + bx + c$.
• Step 2: Simplify the expression based on these substitutions.

Working through these steps:

Step 1: Start with the expression $y = ax^2 + bx + c$.

Since $a = 0$, then $ax^2 = 0 \cdot x^2 = 0$.
Since $b = 1$, then $bx = 1 \cdot x = x$.
Since $c = 0$, then $c = 0$.

Step 2: Plug these values into the equation:

The expression simplifies to:

$y = 0 + x + 0$

Thus, the simplified algebraic expression is $y = x$.

Therefore, the solution to the problem is $x$.

We begin by noting that the general form of a quadratic function is represented by the equation:

$y = ax^2 + bx + c$

Given the parameters $a = -1$, $b = 0$, and $c = 0$, we substitute these values into the equation:

$y = (-1)x^2 + (0)x + 0$

Simplifying the expression, we get:

$y = -x^2$

Thus, the algebraic expression representing the given parameters is $-x^2$.

The correct answer choice that corresponds to this expression is:

$-x^2$

To solve this problem, we will follow these steps:

• Step 1: Identify the given parameters $a = 3$, $b = 6$, $c = 9$.
• Step 2: Use the standard formula for a quadratic expression, which is $y = ax^2 + bx + c$.
• Step 3: Substitute the given values into this formula.

Now, let's work through each step:
Step 1: We have the parameters $a = 3$, $b = 6$, $c = 9$.
Step 2: The standard form of a quadratic equation is $y = ax^2 + bx + c$.
Step 3: Substituting the given values into the expression, we get:

$y = 3x^2 + 6x + 9$

Therefore, the algebraic expression based on the given parameters is:

$3x^2 + 6x + 9$.

To solve this problem, we will construct an algebraic expression using the given parameters in a quadratic function format.

• Step 1: Identify the formula required. The standard quadratic function is given by $ax^2 + bx + c$.
• Step 2: Substitute the given values of $a$, $b$, and $c$. We have $a = 2$, $b = 0$, and $c = 6$.
• Step 3: Insert these values into the formula: $2x^2 + 0x + 6$.
• Step 4: Simplify the expression. Since the coefficient of $x$ is zero, $0x$ can be omitted. This simplifies the expression to $2x^2 + 6$.

The final algebraic expression, representing the given parameters in a quadratic form, is $2x^2 + 6$.

Therefore, the correct algebraic expression is $2x^2 + 6$.

To solve this problem, we will derive the algebraic expression step-by-step:

Step 1: Identify the given information:
The problem states $a = 2$, $b = 0$, and $c = 4$.

Step 2: Write the standard quadratic expression:
The general form is $y = ax^2 + bx + c$.

Step 3: Substitute the given values into the expression:
Replace $a$ with 2, $b$ with 0, and $c$ with 4:
$y = 2x^2 + 0x + 4$.

Step 4: Simplify the expression:
Since $0x$ is zero, the expression simplifies to:
$y = 2x^2 + 4$.

Thus, the algebraic expression based on the given parameters is $2x^2 + 4$.

The correct answer is: $2x^2 + 4$ (Choice 1).

To solve this problem, let's proceed with the construction of the quadratic expression:

• Step 1: Recognize the standard form of a quadratic expression, which is $ax^2 + bx + c$.
• Step 2: Substitute the given values into this formula:
  • $a = 1$
  • $b = 16$
  • $c = 64$
  Plugging in these values, we determine the expression to be $1x^2 + 16x + 64$, which simplifies to $x^2 + 16x + 64$.

Thus, the algebraic expression we derive from these parameters is the quadratic expression:

$x^2 + 16x + 64$

This matches the correct choice provided in the given multiple-choice options.