Which Is The Graph Of The Function F ( X ) = 1 2 X 2 + 2 X − 6 F(x) = \frac{1}{2} X^2 + 2x - 6 F ( X ) = 2 1 ​ X 2 + 2 X − 6 ?

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Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards. In this article, we will explore the graph of the quadratic function $f(x) = \frac{1}{2} x^2 + 2x - 6$.

The Standard Form of a Quadratic Function

The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. In this form, $a$ is the coefficient of the $x^2$ term, $b$ is the coefficient of the $x$ term, and $c$ is the constant term. The graph of a quadratic function in standard form is a parabola that opens upwards if $a > 0$ and downwards if $a < 0$.

Graphing the Quadratic Function $f(x) = \frac{1}{2} x^2 + 2x - 6$

To graph the quadratic function $f(x) = \frac{1}{2} x^2 + 2x - 6$, we need to find the vertex of the parabola. The vertex of a parabola is the point on the parabola that is lowest or highest point. To find the vertex, we can use the formula $x = -\frac{b}{2a}$.

Finding the Vertex

To find the vertex of the parabola, we need to find the value of $x$ that makes the derivative of the function equal to zero. The derivative of the function $f(x) = \frac{1}{2} x^2 + 2x - 6$ is $f'(x) = x + 2$. Setting the derivative equal to zero, we get:

$$x + 2 = 0$$

Solving for $x$, we get:

$$x = -2$$

Now that we have found the value of $x$ that makes the derivative equal to zero, we can find the value of $y$ by plugging this value of $x$ into the original function:

$$f(-2) = \frac{1}{2} (-2)^2 + 2(-2) - 6$$

Simplifying, we get:

$$f(-2) = \frac{1}{2} (4) - 4 - 6$$

$$f(-2) = 2 - 4 - 6$$

$$f(-2) = -8$$

So, the vertex of the parabola is the point $(-2, -8)$.

Finding the Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. To find the axis of symmetry, we need to find the value of $x$ that makes the function equal to zero. Setting the function equal to zero, we get:

$$\frac{1}{2} x^2 + 2x - 6 = 0$$

Multiplying both sides by 2, we get:

$$x^2 + 4x - 12 = 0$$

Factoring the left-hand side, we get:

$$(x + 6)(x - 2) = 0$$

Solving for $x$, we get:

$$x + 6 = 0 \quad \text{or} \quad x - 2 = 0$$

$$x = -6 \quad \text{or} \quad x = 2$$

So, the axis of symmetry is the vertical line $x = -2$.

Graphing the Parabola

Now that we have found the vertex and the axis of symmetry, we can graph the parabola. The parabola opens upwards because the coefficient of the $x^2$ term is positive. The vertex is the point $(-2, -8)$, and the axis of symmetry is the vertical line $x = -2$.

Conclusion

In this article, we have graphed the quadratic function $f(x) = \frac{1}{2} x^2 + 2x - 6$. We have found the vertex of the parabola, which is the point $(-2, -8)$, and the axis of symmetry, which is the vertical line $x = -2$. We have also graphed the parabola, which opens upwards because the coefficient of the $x^2$ term is positive.

Final Answer

The graph of the function $f(x) = \frac{1}{2} x^2 + 2x - 6$ is a parabola that opens upwards. The vertex of the parabola is the point $(-2, -8)$, and the axis of symmetry is the vertical line $x = -2$.

$$

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards. In this article, we will explore the graph of the quadratic function $f(x) = \frac{1}{2} x^2 + 2x - 6$.

Q&A

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point $(-2, -8)$.

Q: What is the axis of symmetry of the parabola?

A: The axis of symmetry of the parabola is the vertical line $x = -2$.

Q: Does the parabola open upwards or downwards?

A: The parabola opens upwards because the coefficient of the $x^2$ term is positive.

Q: How do I find the vertex of the parabola?

A: To find the vertex of the parabola, you need to find the value of $x$ that makes the derivative of the function equal to zero. The derivative of the function $f(x) = \frac{1}{2} x^2 + 2x - 6$ is $f'(x) = x + 2$. Setting the derivative equal to zero, you get:

$$x + 2 = 0$$

Solving for $x$, you get:

$$x = -2$$

Now that you have found the value of $x$ that makes the derivative equal to zero, you can find the value of $y$ by plugging this value of $x$ into the original function:

$$f(-2) = \frac{1}{2} (-2)^2 + 2(-2) - 6$$

Simplifying, you get:

$$f(-2) = \frac{1}{2} (4) - 4 - 6$$

$$f(-2) = 2 - 4 - 6$$

$$f(-2) = -8$$

So, the vertex of the parabola is the point $(-2, -8)$.

Q: How do I find the axis of symmetry of the parabola?

A: To find the axis of symmetry of the parabola, you need to find the value of $x$ that makes the function equal to zero. Setting the function equal to zero, you get:

$$\frac{1}{2} x^2 + 2x - 6 = 0$$

Multiplying both sides by 2, you get:

$$x^2 + 4x - 12 = 0$$

Factoring the left-hand side, you get:

$$(x + 6)(x - 2) = 0$$

Solving for $x$, you get:

$$x + 6 = 0 \quad \text{or} \quad x - 2 = 0$$

$$x = -6 \quad \text{or} \quad x = 2$$

So, the axis of symmetry is the vertical line $x = -2$.

Q: Can I graph the parabola using a calculator?

A: Yes, you can graph the parabola using a calculator. To graph the parabola, you need to enter the function into the calculator and use the graphing feature. You can also use the calculator to find the vertex and the axis of symmetry of the parabola.

Conclusion

In this article, we have explored the graph of the quadratic function $f(x) = \frac{1}{2} x^2 + 2x - 6$. We have found the vertex of the parabola, which is the point $(-2, -8)$, and the axis of symmetry, which is the vertical line $x = -2$. We have also graphed the parabola, which opens upwards because the coefficient of the $x^2$ term is positive.

Final Answer

The graph of the function $f(x) = \frac{1}{2} x^2 + 2x - 6$ is a parabola that opens upwards. The vertex of the parabola is the point $(-2, -8)$, and the axis of symmetry is the vertical line $x = -2$.