Quadratic Function Practice Problems - Parabola Vertex & Graph

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 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.

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

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.