Evaluating the Quadratic Function

Oct 13, 2025 by leolavanderia.com.br 9 views

Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by $p(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this article, we will evaluate the quadratic function $p(x) = -2x^2 + 12x - 19$ for each value of $x$ in the table below.

The Quadratic Function

The quadratic function we will be evaluating is given by $p(x) = -2x^2 + 12x - 19$ . This function has a negative leading coefficient, which means the parabola opens downward. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$ .

Evaluating the Quadratic Function

To evaluate the quadratic function for each value of $x$ in the table, we will substitute each value of $x$ into the function and simplify.

Evaluating $p(1)$

To evaluate $p(1)$ , we substitute $x = 1$ into the function:

p(1) = -2(1)^2 + 12(1) - 19

p(1) = -2 + 12 - 19

p(1) = -9

Evaluating $p(1.5)$

To evaluate $p(1.5)$ , we substitute $x = 1.5$ into the function:

p(1.5) = -2(1.5)^2 + 12(1.5) - 19

p(1.5) = -2(2.25) + 18 - 19

p(1.5) = -4.5 + 18 - 19

p(1.5) = -5.5

Evaluating $p(2)$

To evaluate $p(2)$ , we substitute $x = 2$ into the function:

p(2) = -2(2)^2 + 12(2) - 19

p(2) = -2(4) + 24 - 19

p(2) = -8 + 24 - 19

p(2) = -3

Evaluating $p(2.5)$

To evaluate $p(2.5)$ , we substitute $x = 2.5$ into the function:

p(2.5) = -2(2.5)^2 + 12(2.5) - 19

p(2.5) = -2(6.25) + 30 - 19

p(2.5) = -12.5 + 30 - 19

p(2.5) = -1.5

Evaluating $p(3)$

To evaluate $p(3)$ , we substitute $x = 3$ into the function:

p(3) = -2(3)^2 + 12(3) - 19

p(3) = -2(9) + 36 - 19

p(3) = -18 + 36 - 19

p(3) = -1

Evaluating $p(3.5)$

To evaluate $p(3.5)$ , we substitute $x = 3.5$ into the function:

p(3.5) = -2(3.5)^2 + 12(3.5) - 19

p(3.5) = -2(12.25) + 42 - 19

p(3.5) = -24.5 + 42 - 19

p(3.5) = 1.5

Evaluating $p(4)$

To evaluate $p(4)$ , we substitute $x = 4$ into the function:

p(4) = -2(4)^2 + 12(4) - 19

p(4) = -2(16) + 48 - 19

p(4) = -32 + 48 - 19

p(4) = 3

Conclusion

In this article, we evaluated the quadratic function $p(x) = -2x^2 + 12x - 19$ for each value of $x$ in the table below. We found that the function has a negative leading coefficient, which means the parabola opens downward. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$ . We also found that the function has a maximum value of $3$ at $x = 4$ and a minimum value of $-9$ at $x = 1$ .

Table of Values

$x$	$p(x)$
1	-9
1.5	-5.5
2	-3
2.5	-1.5
3	-1
3.5	1.5
4	3

Discussion

The quadratic function $p(x) = -2x^2 + 12x - 19$ is a polynomial function of degree two. The function has a negative leading coefficient, which means the parabola opens downward. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$ . The function has a maximum value of $3$ at $x = 4$ and a minimum value of $-9$ at $x = 1$ . The table of values shows the values of the function for each value of $x$ in the table.

References

[1] "Quadratic Functions". Math Open Reference. Retrieved 2023-02-20.
[2] "Quadratic Formula". Math Is Fun. Retrieved 2023-02-20.
Quadratic Function Q&A: Frequently Asked Questions =====================================================

Introduction

In our previous article, we evaluated the quadratic function $p(x) = -2x^2 + 12x - 19$ for each value of $x$ in the table below. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is given by $p(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. The vertex can be found using the formula $x = -\frac{b}{2a}$ .

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to find the value of $x$ that makes the derivative of the function equal to zero. The derivative of a quadratic function is given by $p'(x) = 2ax + b$ . Setting the derivative equal to zero and solving for $x$ will give you the value of $x$ at the vertex.

Q: What is the maximum or minimum value of a quadratic function?

A: The maximum or minimum value of a quadratic function is the value of the function at the vertex. If the parabola opens upward, the vertex is a minimum point, and if the parabola opens downward, the vertex is a maximum point.

Q: How do I determine if a quadratic function is increasing or decreasing?

A: To determine if a quadratic function is increasing or decreasing, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the function is increasing, and if the coefficient is negative, the function is decreasing.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is all real numbers. This means that the function is defined for all values of $x$ .

Q: What is the range of a quadratic function?

A: The range of a quadratic function is all real numbers. This means that the function can take on any value.

Q: Can a quadratic function have more than one vertex?

A: No, a quadratic function can only have one vertex.

Q: Can a quadratic function be a linear function?

A: No, a quadratic function cannot be a linear function. A quadratic function has a degree of two, while a linear function has a degree of one.

Q: Can a quadratic function be a constant function?

A: No, a quadratic function cannot be a constant function. A quadratic function has a degree of two, while a constant function has a degree of zero.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We hope that this article has helped you to better understand quadratic functions and how to work with them.

References

[1] "Quadratic Functions". Math Open Reference. Retrieved 2023-02-20.
[2] "Quadratic Formula". Math Is Fun. Retrieved 2023-02-20.

Additional Resources

[1] "Quadratic Functions". Khan Academy. Retrieved 2023-02-20.
[2] "Quadratic Formula". Wolfram MathWorld. Retrieved 2023-02-20.