Ways of Representing the Quadratic Function - 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 create the algebraic expression for the quadratic function given the parameters, we follow these steps:

• Step 1: Identify the values to substitute into the equation. Here, we have $a = -1$, $b = -2$, and $c = -5$.
• Step 2: Use the standard quadratic equation format $y = ax^2 + bx + c$.
• Step 3: Substitute the known values into the equation:

Substituting these values, we get:
$y = (-1)x^2 + (-2)x + (-5)$

Simplify this expression:
This simplifies to $-x^2 - 2x - 5$.

Therefore, the algebraic expression is $-x^2 - 2x - 5$.

To derive the algebraic expression based on the parameters given, we follow these steps:

• Step 1: Recognize the given parameters: $a = 4$, $b = 2$, and $c = 5$.
• Step 2: Acknowledge that the standard form for a quadratic expression is $y = ax^2 + bx + c$.
• Step 3: Substitute the given parameter values into this quadratic expression.

Now, let's implement these steps to form the quadratic expression:
Step 1: The given parameters are $a = 4$, $b = 2$, and $c = 5$.
Step 2: Our basis is the quadratic form $y = ax^2 + bx + c$.
Step 3: Substituting the given values, we find:

$y = 4x^2 + 2x + 5$

This substitution provides us with the quadratic expression $y = 4x^2 + 2x + 5$, fulfilling the problem's requirements.

Therefore, the correct algebraic expression is $4x^2 + 2x + 5$.

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 determine the algebraic expression, we will substitute the given parameters into the standard form of the quadratic function:

• The standard quadratic form is $y = ax^2 + bx + c$.
• Substitute $a = 1$, $b = 1$, and $c = 0$ into the equation.

Substituting these values, the expression becomes:

$y = 1 \cdot x^2 + 1 \cdot x + 0$.

This simplifies to:

$y = x^2 + x$.

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

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, 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.

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 solve this problem, we'll follow these steps:

• Step 1: Identify and substitute the values of $a$, $b$, and $c$ into the equation $y = ax^2 + bx + c$.
• Step 2: Simplify the equation to obtain the required expression.
• Step 3: Compare the simplified expression with the provided multiple-choice answers.

Let's work through each step:

Step 1: The given coefficients are $a = \frac{1}{2}$, $b = \frac{1}{2}$, and $c = \frac{1}{2}$. Substitute these values into the standard quadratic form $y = ax^2 + bx + c$:

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

Step 2: The expression is already simplified. The coefficients are correctly substituted, and no further simplification is needed:

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

Step 3: Compare this expression to the provided multiple-choice options. The correct match is:

Choice 1: $\frac{x^2}{2} + \frac{x}{2} + \frac{1}{2}$

Therefore, the algebraic expression is $\frac{x^2}{2} + \frac{x}{2} + \frac{1}{2}$.

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

• Step 1: Identify the general form of the quadratic expression.
• Step 2: Substitute the given values $a = 5$, $b = 3$, and $c = -4$ into the quadratic form.
• Step 3: Write down the resultant expression.

Now, let's work through each step:
Step 1: The general form of a quadratic expression is $ax^2 + bx + c$.
Step 2: We are given $a = 5$, $b = 3$, and $c = -4$. Substituting these into the expression, we get:

$5x^2 + 3x - 4$

Therefore, the solution to the problem is $5x^2 + 3x - 4$.

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

• Step 1: Identify the given information
• Step 2: Apply the standard quadratic formula
• Step 3: Perform the substitution

Now, let's work through each step:
Step 1: We are given $a = 1$, $b = 2$, and $c = 0$.
Step 2: We'll use the formula $y = ax^2 + bx + c$ to form our expression.
Step 3: By substituting the given values, we get:

• Replace $a$ with 1: $1x^2 = x^2$.
• Replace $b$ with 2: $2x = 2x$.
• Replace $c$ with 0: since $c = 0$, it does not contribute to the expression, so we can omit this term.

Therefore, we combine the terms to form the expression: $x^2 + 2x$.

The correct answer choice based on our derived expression is: $x^2 + 2x$.

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

• Step 1: Identify the provided coefficients for the expression of the form $ax^2 + bx + c$.
• Step 2: Substitute the provided values into the expression.

Now, let's perform these steps:

Step 1: The problem provides us with the coefficients $a = 1$, $b = -1$, and $c = 3$ for a quadratic expression $ax^2 + bx + c$.

Step 2: Substitute these values into the quadratic expression:

$a = 1$: Multiply $x^2$ by $1$, resulting in $x^2$.

$b = -1$: Multiply $x$ by $-1$, resulting in $-x$.

$c = 3$: The constant term is $3$.

Thus, the algebraic expression is:

$x^2 - x + 3$.

Comparing this result to the given choices, we find that this expression matches choice 3.

Therefore, the solution to the problem is $x^2 - x + 3$.

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'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 $2x^2 + \frac{1}{2}x + 4$.