Orientation of the Parabola

Oct 13, 2025 by recoveryshake.com 146 views

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of quadratic functions and analyze the given function $f(x) = -2(x-3)(x+1)$ .

Standard Form

The standard form of a quadratic function is $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. To convert the given function to standard form, we need to expand the expression.

Expanding the Expression

$f(x) = -2(x-3)(x+1)$

$f(x) = -2(x^2 + x - 3x - 3)$

$f(x) = -2(x^2 - 2x - 3)$

$f(x) = -2x^2 + 4x + 6$

Identifying Coefficients

Now that we have the function in standard form, we can identify the coefficients $a$ , $b$ , and $c$ .

a: $-2$
b: $4$
c: $6$

Orientation of the Parabola

The coefficient $a$ determines the orientation of the parabola. If $a > 0$ , the parabola opens upwards, and if $a < 0$ , the parabola opens downwards.

Orientation: The parabola opens downwards since $a = -2$ .

Vertex

The vertex of a quadratic function is the point at which the parabola changes direction. It can be found using the formula $x = -\frac{b}{2a}$ .

Vertex: $x = -\frac{4}{2(-2)} = 1$

To find the y-coordinate of the vertex, we substitute the x-coordinate into the function.

y-coordinate: $f(1) = -2(1)^2 + 4(1) + 6 = 8$

Therefore, the vertex is $(1, 8)$ .

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. It can be found using the formula $x = -\frac{b}{2a}$ .

Axis of Symmetry: $x = 1$

y-intercept

The y-intercept is the point at which the parabola intersects the y-axis. It can be found by substituting $x = 0$ into the function.

y-intercept: $f(0) = -2(0)^2 + 4(0) + 6 = 6$

x-intercept(s)

The x-intercepts are the points at which the parabola intersects the x-axis. They can be found by setting the function equal to zero and solving for $x$ .

x-intercepts: $-2x^2 + 4x + 6 = 0$

Using the quadratic formula, we get:

$x = \frac{-4 \pm \sqrt{16 - 4(-2)(6)}}{2(-2)}$

$x = \frac{-4 \pm \sqrt{16 + 48}}{-4}$

$x = \frac{-4 \pm \sqrt{64}}{-4}$

$x = \frac{-4 \pm 8}{-4}$

Therefore, the x-intercepts are $x = -3$ and $x = 1$ .

Conclusion

In conclusion, we have analyzed the given quadratic function $f(x) = -2(x-3)(x+1)$ and filled in the table with the required information.

Property	Value
a	$-2$
b	$4$
c	$6$
Orientation	Downwards
Vertex	$(1, 8)$
Axis of Symmetry	$x = 1$
y-intercept	$6$
x-intercept(s)	$-3, 1$

Introduction

In our previous article, we analyzed the quadratic function $f(x) = -2(x-3)(x+1)$ and filled in the table with the required information. In this article, we will answer some frequently asked questions about quadratic functions.

Q&A

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. It can be written in the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I convert a quadratic function to standard form?

A: To convert a quadratic function to standard form, you need to expand the expression. For example, if the function is $f(x) = -2(x-3)(x+1)$ , you can expand it as follows:

$f(x) = -2(x^2 + x - 3x - 3)$

$f(x) = -2(x^2 - 2x - 3)$

$f(x) = -2x^2 + 4x + 6$

Q: What is the coefficient $a$ in a quadratic function?

A: The coefficient $a$ determines the orientation of the parabola. If $a > 0$ , the parabola opens upwards, and if $a < 0$ , the parabola opens downwards.

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

A: The vertex of a quadratic function is the point at which the parabola changes direction. It can be found using the formula $x = -\frac{b}{2a}$ .

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

A: The axis of symmetry is a vertical line that passes through the vertex. It can be found using the formula $x = -\frac{b}{2a}$ .

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

A: The y-intercept is the point at which the parabola intersects the y-axis. It can be found by substituting $x = 0$ into the function.

Q: How do I find the x-intercept(s) of a quadratic function?

A: The x-intercepts are the points at which the parabola intersects the x-axis. They can be found by setting the function equal to zero and solving for $x$ .

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. In other words, a quadratic function has a squared term, while a linear function does not.

Q: Can a quadratic function have more than one x-intercept?

A: Yes, a quadratic function can have more than one x-intercept. This occurs when the function has two real roots.

Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts. This occurs when the function has no real roots.

Conclusion

In conclusion, we have answered some frequently asked questions about quadratic functions. By understanding the properties of quadratic functions, we can solve various mathematical problems and gain a deeper insight into the world of mathematics.

Quadratic Function Formula

The quadratic function formula is:

$f(x) = ax^2 + bx + c$

where $a$ , $b$ , and $c$ are constants.

Quadratic Function Graph

The quadratic function graph is a parabola that opens upwards or downwards, depending on the value of $a$ .

Quadratic Function Examples

$f(x) = x^2 + 2x + 1$
$f(x) = -2x^2 + 4x - 3$
$f(x) = x^2 - 4x + 4$

By understanding the properties of quadratic functions, we can solve various mathematical problems and gain a deeper insight into the world of mathematics.