Quadratic Function Practice Problems - Parabola Vertex & Graph

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.

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:

• 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 $a = 2$, $b = -3$, $c = -6$.

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 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 $a = -3, b = 0, c = -4$.