The Quadratic Function - Examples, Exercises and Solutions

To solve this problem, we will identify the parameters of the given quadratic function step-by-step:

• Step 1: Define the problem statement: We have $y = x^2 + x + 5$.
• Step 2: Identify the standard form of a quadratic function, which is $y = ax^2 + bx + c$.
• Step 3: Compare the given quadratic expression with the standard form to identify the coefficients.

Now, let's analyze the quadratic function provided:

From the given expression $y = x^2 + x + 5$:
- The coefficient of $x^2$ is $1$, so $a = 1$.
- The coefficient of $x$ is $1$, so $b = 1$.
- The constant term is $5$, so $c = 5$.

Therefore, the parameters of the quadratic function are $a = 1$, $b = 1$, and $c = 5$.

Consequently, the correct choice from the provided options is $(a = 1, b = 1, c = 5)$.

To solve the problem of identifying the coefficients in the quadratic function $y = -x^2 + x + 5$, we follow these steps:

• Step 1: Write down the general form of a quadratic equation: $y = ax^2 + bx + c$.

• Step 2: Compare the given equation $y = -x^2 + x + 5$ to the general form.

• Step 3: Identify the value of each coefficient:

  • The coefficient of $x^2$ is $-1$, so $a = -1$.

  • The coefficient of $x$ is $+1$, so $b = 1$.

  • The constant term is $+5$, so $c = 5$.

Therefore, the parameters of the quadratic function are $a = -1$, $b = 1$, and $c = 5$.

This matches choice 2, confirming the parameters in the quadratic function.

Final Answer: $a=-1, b=1, c=5$.

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

• Step 1: Compare the given quadratic function with the standard form.
• Step 2: Directly identify the coefficients $a$, $b$, and $c$.
• Step 3: Verify the correct choice from the provided options, if applicable.

Now, let's work through each step:
Step 1: The given quadratic function is $y = 3x^2 + 3x - 4$. The standard form for a quadratic equation is $y = ax^2 + bx + c$.
Step 2: By comparing the given equation to the standard form, we can identify the coefficients:
- $a = 3$, from the term $3x^2$.
- $b = 3$, from the term $3x$.
- $c = -4$, from the constant term $-4$.
Step 3: With these values, compare them to the given choices. The choice that matches these values is option 3: $a = 3, b = 3, c = -4$.

Therefore, the solution to the problem is $a = 3, b = 3, c = -4$.

To solve this problem, we'll compare the given quadratic function with its standard form:

• Step 1: Recognize the given function as $y = -4x^2 + 3$.
• Step 2: Write down the standard form of a quadratic function, which is $y = ax^2 + bx + c$.
• Step 3: Match corresponding terms to identify $a$, $b$, and $c$.

Now, let's work through these steps:

Step 1: The given function is $y = -4x^2 + 3$.

Step 2: The standard form of a quadratic function is $y = ax^2 + bx + c$.

Step 3: By direct comparison:
- The coefficient of $x^2$ in the given expression is $-4$. Therefore, $a = -4$.
- There is no $x$ term in the given expression, which implies the coefficient $b = 0$.
- The constant term in the given expression is $3$, indicating $c = 3$.

Therefore, the solution is $a = -4$, $b = 0$, $c = 3$, which matches with choice 3.

To solve this problem, we will follow these steps:

• Step 1: Identify the form of a standard quadratic equation.
• Step 2: Compare the given function with the quadratic standard form.
• Step 3: Match the coefficients to the given answer choices.

Now, let's work through each step:
Step 1: The standard form of a quadratic function is $y = ax^2 + bx + c$.
Step 2: Given the function $y = -3x^2 - 4$, we compare this with the standard form:

• Coefficient $a$ is associated with $x^2$. Here, $a = -3$.
• Coefficient $b$ is associated with $x$. Since there is no $x$ term, $b = 0$.
• The constant term $c$ is the standalone number, which is $c = -4$.

Therefore, the solution to the problem is $a = -3, b = 0, c = -4$.

To solve the problem, we'll identify the parameters $a$, $b$, and $c$ from the given quadratic function:

• **Step 1**: Compare the given equation $y = -5x^2$ with the general quadratic form $y = ax^2 + bx + c$.
• **Step 2**: Note that the coefficient of $x^2$ is $-5$, hence $a = -5$.
• **Step 3**: Since there is no $x$ term, $b = 0$.
• **Step 4**: Since there is no constant term, $c = 0$.

After identifying the parameters, we conclude:

The parameters for the quadratic function are $a = -5$, $b = 0$, $c = 0$. Therefore, the correct choice is:

The correct answer is $a = -5, b = 0, c = 0$.

To solve this problem, we will follow these steps:

• Step 1: Identify the given quadratic equation and its form.
• Step 2: Directly match the coefficients of the given equation to the standard form $y = ax^2 + bx + c$.
• Step 3: Compare with the provided choices and select the one that matches.

Now, let's work through each step:
Step 1: The given quadratic equation is $y = -x^2 + 3x + 40$. This matches the form $y = ax^2 + bx + c$.

Step 2: By comparing the given equation to the standard form:
- The coefficient $a$ is the coefficient of $x^2$, which is $-1$.
- The coefficient $b$ is the coefficient of $x$, which is $3$.
- The coefficient $c$ is the constant term, which is $40$.

Step 3: From the analysis, we identify $a = -1$, $b = 3$, $c = 40$. We compare these with the provided choices.
The correct answer is: $a=-1,b=3,c=40$

Therefore, the solution to the problem matches choice 4.

To solve this problem, we will identify values of $a$, $b$, and $c$ in the quadratic function:

• Step 1: Note the given equation $y = 3x^2 - 81$.
• Step 2: Compare it to the standard quadratic form $y = ax^2 + bx + c$.
• Step 3: Match coefficients to find $a$, $b$, and $c$.

Now, let's work through each step:
Step 1: The given equation is $y = 3x^2 - 81$.
Step 2: Compare this to the standard form, $y = ax^2 + bx + c$. In this equation:
- The coefficient of $x^2$ is 3, hence $a = 3$.
- There is no $x$ term, which means $b = 0$.
- The constant term is $-81$, hence $c = -81$.

Therefore, the solution to the problem is $a = 3, b = 0, c = -81$.

To solve this problem, let's follow these steps:

• Step 1: Recognize the standard form of a quadratic equation.
• Step 2: Match the given function to the standard form.
• Step 3: Identify each coefficient $a$, $b$, and $c$.

Now, let's work through these steps:

Step 1: The standard form of a quadratic function is $y = ax^2 + bx + c$. Our goal is to identify $a$, $b$, and $c$.

Step 2: We are given the function $y = x^2$. This can be aligned with the standard form as $y = 1 \cdot x^2 + 0 \cdot x + 0$.

Step 3: By comparing the given function $y = x^2$ with the standard form, we can deduce:
- The coefficient of $x^2$ is 1, so $a = 1$.
- The linear term coefficient is missing, which implies $b = 0$.
- There is no constant term, so $c = 0$.

Therefore, the coefficients are $a = 1, b = 0, c = 0$, corresponding to choice 1.

To solve this problem, we will follow these steps:

• Step 1: Identify each term in the given function $y = 2x^2 + 3$.
• Step 2: Compare the equation to the standard quadratic form $y = ax^2 + bx + c$.
• Step 3: Determine the coefficients $a$, $b$, and $c$.
• Step 4: Match these coefficients to the correct multiple-choice option.

Step 1: The given function is $y = 2x^2 + 3$. There is no $x$ term present.

Step 2: Compare this with the standard form $y = ax^2 + bx + c$:

• The coefficient of $x^2$ is $a = 2$.
• The coefficient of $x$ is $b = 0$ because there is no $x$ term.
• The constant term is $c = 3$.

Step 3: Therefore, the coefficients are $a = 2$, $b = 0$, and $c = 3$.

Step 4: Review the multiple-choice options provided:

• Choice 1: $a = 0$, $b = 2$, $c = 3$
• Choice 2: $a = 0$, $b = 3$, $c = 2$
• Choice 3: $a = 2$, $b = 0$, $c = 3$
• Choice 4: $a = 3$, $b = 0$, $c = 2$

The correct choice is Choice 3: $a = 2$, $b = 0$, $c = 3$.

Therefore, the solution to the problem is the values $a = 2$, $b = 0$, $c = 3$ which correspond to choice 3.

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

$c = 0$

a is the coefficient of X², here it does not have a coefficient, therefore

$a = 1$

$b= 10$

is the number that comes before the X that is not squared.

To solve this problem, we'll clearly delineate the given expression and compare it to the standard quadratic form:

• Step 1: Recognize the standard form of a quadratic equation as $y = ax^2 + bx + c$.
• Step 2: Compare the given equation $y = x^2 - 6x + 4$ to the standard form.
• Step 3: Identify coefficients:
  - The coefficient of $x^2$ is $a = 1$.
  - The coefficient of $x$ is $b = -6$.
  - The constant term is $c = 4$.

Therefore, the coefficients for the quadratic function $y = x^2 - 6x + 4$ are $a = 1$, $b = -6$, and $c = 4$.

Among the provided choices, choice 3: $a=1,b=-6,c=4$ is the correct one.

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

That is,

a is the coefficient of x², in this case 2.
b is the coefficient of x, in this case 5.
And c is the number without a variable at the end, in this case 6.

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

• Identify the given quadratic function.
• Match it with the standard form of a quadratic equation $y = ax^2 + bx + c$.
• Extract the values of $a$, $b$, and $c$ directly from the comparison.

Now, let's work through each step:
Step 1: The given quadratic function is $y = 2x^2 - 3x - 6$.
Step 2: The standard form of a quadratic equation is $y = ax^2 + bx + c$.
Step 3: By matching the given quadratic function with the standard form:

- The coefficient of $x^2$ is $2$, so $a = 2$.
- The coefficient of $x$ is $-3$, so $b = -3$.
- The constant term is $-6$, so $c = -6$.

Therefore, the solution to the problem is $a = 2$, $b = -3$, $c = -6$.

The given quadratic function is $y = 5x^2 - 4x - 30$.
To identify the coefficients, let's compare this with the standard form of a quadratic equation, $y = ax^2 + bx + c$.

• Step 1: Compare the terms of the given equation to the standard form.
• Step 2: Identify each coefficient:
  For $ax^2$, $a = 5$ in the term $5x^2$.
  For $bx$, $b = -4$ in the term $-4x$.
  For the constant term $c$, $c = -30$.

Thus, we have identified the coefficients as $a = 5$, $b = -4$, and $c = -30$.

Therefore, the correct answer is $a = 5, b = -4, c = -30$.

The correct choice is .