Plotting Functions with Parameters - Examples, Exercises and Solutions

The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:

The question asked in the problem - What is the value of the coefficient$a$ in the equation?

Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:

The rule states that the roots of an equation of the form:

$ax^2+bx+c=0$are:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient $a$is the coefficient of the quadratic term (meaning the term with the second power)- $x^2$Let's examine the equation in the problem:

$$$-x^2+7x-9 =0$

Remember that the minus sign before the quadratic term means multiplication by: $-1$ , therefore- we can write the equation as:

$-1\cdot x^2+7x-9 =0$

The number that multiplies the $x^2$, is $-1$ hence we identify that the coefficient of the quadratic term is the number $-1$,

Therefore the correct answer is A.

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

In the problem, the question was asked: what is the value of the coefficient$b$in the equation?

Let's remember the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$are :

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient$b$is the coefficient of the term in the first power -$x$We then examine the equation of the given problem:

$$$3x^2+8x-5 =0$That is, the number that multiplies

$x$ is

$8$Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number$8$,

Thus the correct answer is option d.

The quadratic equation is given as $4x^2 + 9x - 2$. This equation is in the standard form of a quadratic equation, which is $ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients.

• The term $4x^2$ indicates that the coefficient $a = 4$.
• The term $9x$ indicates that the coefficient $b = 9$.
• The constant term $-2$ indicates that the coefficient $c = -2$.

From this analysis, we can see that the coefficient $c$ is $-2$.

Therefore, the value of the coefficient $c$ in the equation is $-2$.

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.

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 $a = 2$, $b = -3$, $c = -6$.

Let's determine the coefficients for the quadratic function given by $y = -2x^2 + 3x + 10$.

• Step 1: Identify $a$.
  The coefficient of $x^2$ is $-2$. Thus, $a = -2$.
• Step 2: Identify $b$.
  The coefficient of $x$ is $3$. Thus, $b = 3$.
• Step 3: Identify $c$.
  The constant term is $10$. Thus, $c = 10$.

Comparing these coefficients to the provided choices, the correct answer is:

$a = -2, b = 3, c = 10$ .

Therefore, the correct choice is Choice 4.

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

• Step 1: Identify the given quadratic function.
• Step 2: Compare it to the standard form $y = ax^2 + bx + c$.
• Step 3: Determine the values of $a$, $b$, and $c$.

Now, let's work through each step:

Step 1: The problem gives us the quadratic function $y = 3x^2 + 4x + 5$.

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

Step 3: By comparing $y = 3x^2 + 4x + 5$ with $y = ax^2 + bx + c$, we find:
- The coefficient of $x^2$ is $a = 3$.
- The coefficient of $x$ is $b = 4$.
- The constant term is $c = 5$.

Therefore, the solution to the problem is $a = 3, b = 4, c = 5$.

This matches choice 2, which states: $a = 3, b = 4, c = 5$.

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

In the problem, the question was asked: what is the value of the coefficient$c$in the equation?

Let's remember the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$are:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient
$c$is the free term - and as such the coefficient of the term is raised to the power of zero -$x^0$(Any number other than zero raised to the power of zero equals 1:

$x^0=1$)

Next we examine the equation of the given problem:

$$$3x^2+5x=0$Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

$3x^2+5x+0=0$and therefore the value of the coefficient$c$ is 0.

Hence the correct answer is option c.

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 will match the given quadratic equation with its standard form:

• Step 1: Identify the standard quadratic form as $y = ax^2 + bx + c$.
• Step 2: Compare with given equation $y = 4 + 3x^2 - x$.
• Step 3: Determine the coefficients by arranging the equation in standard form.

Let's now perform the steps:

Step 1: The standard quadratic form is $y = ax^2 + bx + c$.

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

Step 3: Rearrange the given equation to match the standard form:

$y = 3x^2 - x + 4$.

Now, directly compare:

$a = 3$

$b = -1$

$c = 4$

Therefore, the coefficients are correctly identified as $a=3, b=-1,$ and $c=4$.

The correct answer is: $a=3, b=-1, c=4$.

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

• Step 1: Rearrange the given quadratic function into the standard form.
• Step 2: Identify the coefficients by comparing them with the standard quadratic function.

Now let's work through each step:

Step 1: The given quadratic is $y = 3x^2 + 4 - 5x$. Rearrange this function to align terms with their degrees:
$y = 3x^2 - 5x + 4$.

Step 2: Compare this with the standard quadratic form $y = ax^2 + bx + c$, where:
$a = 3$ (the coefficient of $x^2$),
$b = -5$ (the coefficient of $x$),
$c = 4$ (the constant term).

Therefore, the correct choice is $a = 3, b = -5, c = 4$.