Analyzing the Pairs of Values

Oct 13, 2025 by windailys.com 2 views

Introduction

In mathematics, quadratic functions are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will explore one quadratic function described by the equation $f(x) = 3x^2 + 12x - 36$ and its pairs of values given in a table.

The Quadratic Function

The quadratic function $f(x) = 3x^2 + 12x - 36$ is a polynomial function of degree two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this case, $a = 3$ , $b = 12$ , and $c = -36$ . The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Pairs of Values

The pairs of values for $g(x)$ are given in the table below:

$x$	$g(x)$
-6	0
-4	-12
-2
0
2
4

Analyzing the Pairs of Values

To analyze the pairs of values, we need to find the corresponding values of $g(x)$ for each value of $x$ . We can do this by substituting each value of $x$ into the equation $g(x) = \frac{f(x) - 36}{3}$ .

$x$	$f(x)$	$g(x)$
-6	108	0
-4	60	-12
-2	12
0	-36
2	-24
4	0

Finding the Missing Values

To find the missing values of $g(x)$ , we need to substitute the corresponding values of $x$ into the equation $g(x) = \frac{f(x) - 36}{3}$ .

$x$	$f(x)$	$g(x)$
-6	108	0
-4	60	-12
-2	12	2
0	-36	-6
2	-24	-4
4	0	0

Conclusion

In this article, we explored one quadratic function described by the equation $f(x) = 3x^2 + 12x - 36$ and its pairs of values given in a table. We analyzed the pairs of values and found the missing values of $g(x)$ by substituting the corresponding values of $x$ into the equation $g(x) = \frac{f(x) - 36}{3}$ . This article demonstrates the importance of quadratic functions in mathematics and how they can be used to solve problems.

The Graph of the Quadratic Function

The graph of the quadratic function $f(x) = 3x^2 + 12x - 36$ is a parabola that opens upwards. The vertex of the parabola is at the point $(x, y) = (-2, -12)$ . The axis of symmetry of the parabola is the vertical line $x = -2$ .

The X-Intercepts

The x-intercepts of the parabola are the points where the graph of the quadratic function intersects the x-axis. To find the x-intercepts, we need to set $f(x) = 0$ and solve for $x$ .

$3x^2 + 12x - 36 = 0$

We can factor the quadratic expression as follows:

$(3x - 6)(x + 6) = 0$

This gives us two possible solutions:

$3x - 6 = 0$ or $x + 6 = 0$

Solving for $x$ , we get:

$x = 2$ or $x = -6$

Therefore, the x-intercepts of the parabola are the points $(2, 0)$ and $(-6, 0)$ .

The Y-Intercept

The y-intercept of the parabola is the point where the graph of the quadratic function intersects the y-axis. To find the y-intercept, we need to set $x = 0$ and solve for $y$ .

$f(0) = 3(0)^2 + 12(0) - 36$

$= -36$

Therefore, the y-intercept of the parabola is the point $(0, -36)$ .

The Vertex

The vertex of the parabola is the point where the graph of the quadratic function changes direction. To find the vertex, we need to find the x-coordinate of the vertex, which is given by the formula:

$x = -\frac{b}{2a}$

In this case, $a = 3$ and $b = 12$ , so:

$x = -\frac{12}{2(3)}$

$= -2$

Therefore, the x-coordinate of the vertex is $-2$ . To find the y-coordinate of the vertex, we need to substitute $x = -2$ into the equation $f(x) = 3x^2 + 12x - 36$ .

$f(-2) = 3(-2)^2 + 12(-2) - 36$

$= -12$

Therefore, the vertex of the parabola is the point $(-2, -12)$ .

The Axis of Symmetry

The axis of symmetry of the parabola is the vertical line that passes through the vertex. To find the equation of the axis of symmetry, we need to use the formula:

$x = -\frac{b}{2a}$

In this case, $a = 3$ and $b = 12$ , so:

$x = -\frac{12}{2(3)}$

$= -2$

Therefore, the equation of the axis of symmetry is $x = -2$ .

The Domain and Range

The domain of the quadratic function is the set of all possible input values for $x$ . In this case, the domain is all real numbers, since there are no restrictions on the values of $x$ .

The range of the quadratic function is the set of all possible output values for $y$ . In this case, the range is all real numbers greater than or equal to $-36$ , since the minimum value of the function is $-36$ .

Conclusion

Frequently Asked Questions

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 $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$ .

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you need to set $f(x) = 0$ and solve for $x$ . This will give you the points where the graph of the function intersects the x-axis.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you need to set $x = 0$ and solve for $y$ . This will give you the point where the graph of the function intersects the y-axis.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the graph of the function changes direction. It is the minimum or maximum point of the parabola.

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 x-coordinate of the vertex, which is given by the formula:

$x = -\frac{b}{2a}$

Then, you need to substitute this value of $x$ into the equation $f(x) = ax^2 + bx + c$ to find the y-coordinate of the vertex.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. It is the line of symmetry of the parabola.

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

A: To find the axis of symmetry of a quadratic function, you need to use the formula:

$x = -\frac{b}{2a}$

This will give you the equation of the axis of symmetry.

Q: What is the domain and range of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values for $x$ . The range of a quadratic function is the set of all possible output values for $y$ .

Q: How do I find the domain and range of a quadratic function?

A: To find the domain and range of a quadratic function, you need to analyze the graph of the function. The domain is all real numbers, and the range is all real numbers greater than or equal to the minimum value of the function.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We covered topics such as the graph of a quadratic function, x-intercepts, y-intercept, vertex, axis of symmetry, domain, and range. This article demonstrates the importance of quadratic functions in mathematics and how they can be used to solve problems.

Additional Resources

For more information on quadratic functions, you can check out the following resources:

Khan Academy: Quadratic Functions
Math Is Fun: Quadratic Functions
Wolfram MathWorld: Quadratic Function

Practice Problems

To practice what you have learned, try solving the following problems:

Find the x-intercepts of the quadratic function $f(x) = x^2 + 4x + 4$ .
Find the y-intercept of the quadratic function $f(x) = 2x^2 - 3x - 1$ .
Find the vertex of the quadratic function $f(x) = x^2 - 6x + 8$ .
Find the axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 3$ .
Find the domain and range of the quadratic function $f(x) = x^2 - 4x + 4$ .

Conclusion

In this article, we explored quadratic functions and answered some frequently asked questions. We covered topics such as the graph of a quadratic function, x-intercepts, y-intercept, vertex, axis of symmetry, domain, and range. This article demonstrates the importance of quadratic functions in mathematics and how they can be used to solve problems.