Consider The Following Quadratic Function: F ( X ) = − 3 X 2 − 6 X + 2 F(x) = -3x^2 - 6x + 2 F ( X ) = − 3 X 2 − 6 X + 2 (a) Write The Equation In The Form F ( X ) = A ( X − H ) 2 + K F(x) = A(x-h)^2 + K F ( X ) = A ( X − H ) 2 + K . Then, Give The Vertex Of Its Graph.Writing In The Form Specified: F ( X ) = □ F(x) = \square F ( X ) = □ Vertex:

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Understanding Quadratic Functions

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Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. 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.

Converting Quadratic Functions to Vertex Form

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The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. To convert a quadratic function from the general form to the vertex form, we need to complete the square. This involves manipulating the expression to create a perfect square trinomial.

Converting the Given Quadratic Function

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The given quadratic function is $f(x) = -3x^2 - 6x + 2$. To convert this function to the vertex form, we need to complete the square.

Step 1: Factor out the coefficient of $x^2$

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The coefficient of $x^2$ is $-3$. We can factor this out to get:

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

Step 2: Add and subtract the square of half the coefficient of $x$

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The coefficient of $x$ is $2$. Half of this is $1$, and the square of $1$ is $1$. We can add and subtract $1$ inside the parentheses to get:

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

Step 3: Simplify the expression

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We can simplify the expression by combining the like terms:

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

$$f(x) = -3(x + 1)^2 + 5$$

Vertex of the Graph

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The vertex form of the quadratic function is $f(x) = -3(x + 1)^2 + 5$. The vertex of the graph is the point $(h,k)$, where $h$ is the value that makes the expression inside the parentheses equal to zero, and $k$ is the constant term.

In this case, the expression inside the parentheses is $x + 1$. Setting this equal to zero, we get:

$$x + 1 = 0$$

$$x = -1$$

So, the value of $h$ is $-1$. The value of $k$ is the constant term, which is $5$.

Therefore, the vertex of the graph is the point $(-1,5)$.

Conclusion

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In this article, we have seen how to convert a quadratic function from the general form to the vertex form. We have also seen how to find the vertex of the graph of a quadratic function. The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. We have used the given quadratic function $f(x) = -3x^2 - 6x + 2$ as an example to demonstrate the process of converting a quadratic function to the vertex form and finding the vertex of its graph.

Example Problems

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Problem 1

Convert the quadratic function $f(x) = 2x^2 + 4x - 3$ to the vertex form.

Solution

To convert the quadratic function to the vertex form, we need to complete the square.

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

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

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

$$f(x) = 2(x + 1)^2 - 5$$

The vertex of the graph is the point $(-1,-5)$.

Problem 2

Find the vertex of the graph of the quadratic function $f(x) = -2x^2 + 4x + 1$.

Solution

To find the vertex of the graph, we need to convert the quadratic function to the vertex form.

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

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

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

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

The vertex of the graph is the point $(1,3)$.

Applications of Quadratic Functions

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Quadratic functions have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic functions include:

• Projectile motion: Quadratic functions are used to model the trajectory of a projectile under the influence of gravity.
• Optimization: Quadratic functions are used to optimize functions, such as finding the maximum or minimum value of a function.
• Economics: Quadratic functions are used to model the behavior of economic systems, such as the supply and demand curves.
• Physics: Quadratic functions are used to model the motion of objects, such as the trajectory of a thrown ball.

Conclusion

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In this article, we have seen how to convert a quadratic function from the general form to the vertex form and how to find the vertex of the graph of a quadratic function. We have also seen some of the applications of quadratic functions in various fields. Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields.

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Frequently Asked Questions

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Q: What is a quadratic function?

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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 vertex form of a quadratic function?

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A: The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola.

Q: How do I convert a quadratic function from the general form to the vertex form?

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A: To convert a quadratic function from the general form to the vertex form, you need to complete the square. This involves manipulating the expression to create a perfect square trinomial.

Q: What is the significance of the vertex of a quadratic function?

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A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum value of the function.

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

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A: To find the vertex of a quadratic function, you need to convert the function to the vertex form. The vertex is then given by the point $(h,k)$, where $h$ is the value that makes the expression inside the parentheses equal to zero, and $k$ is the constant term.

Q: What are some common applications of quadratic functions?

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A: Quadratic functions have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic functions include:

• Projectile motion: Quadratic functions are used to model the trajectory of a projectile under the influence of gravity.
• Optimization: Quadratic functions are used to optimize functions, such as finding the maximum or minimum value of a function.
• Economics: Quadratic functions are used to model the behavior of economic systems, such as the supply and demand curves.
• Physics: Quadratic functions are used to model the motion of objects, such as the trajectory of a thrown ball.

Q: How do I determine the direction of the parabola?

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A: To determine the direction of the parabola, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

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

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A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for $x$. The x-intercepts are the points where the graph of the function crosses the x-axis.

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

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A: To find the y-intercept of a quadratic function, you need to evaluate the function at $x=0$. The y-intercept is the point where the graph of the function crosses the y-axis.

Additional Resources

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• Quadratic Function Calculator: A calculator that can help you convert a quadratic function from the general form to the vertex form and find the vertex of the graph.
• Quadratic Function Grapher: A grapher that can help you visualize the graph of a quadratic function and find the x-intercepts and y-intercept.
• Quadratic Function Solver: A solver that can help you solve quadratic equations and find the solutions.

Conclusion

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In this article, we have seen some of the frequently asked questions about quadratic functions. We have also seen some of the common applications of quadratic functions and how to determine the direction of the parabola, find the x-intercepts and y-intercept, and convert a quadratic function from the general form to the vertex form. Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields.