Consider The Quadratic Function F ( X ) = − 2 X 2 + 5 X − 4 F(x)=-2x^2+5x-4 F ( X ) = − 2 X 2 + 5 X − 4 .The Leading Coefficient Of The Function Is − 2 -2 − 2 .

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Introduction to 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, and $a$ is not equal to zero. The leading coefficient of a quadratic function is the coefficient of the term with the highest degree, which in this case is $-2$. In this article, we will explore the properties of the quadratic function $f(x)=-2x^2+5x-4$ and its leading coefficient.

The Leading Coefficient of a Quadratic Function

The leading coefficient of a quadratic function is a crucial part of the function's definition. It determines the direction and the rate at which the function opens or closes. If the leading coefficient is positive, the function opens upwards, and if it is negative, the function opens downwards. In the case of the function $f(x)=-2x^2+5x-4$, the leading coefficient is $-2$, which means that the function opens downwards.

Graphing the Quadratic Function

To graph the quadratic function $f(x)=-2x^2+5x-4$, we need to find the vertex of the parabola. The vertex of a parabola is the point at which the function changes direction. To find the vertex, we can use the formula $x=-\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a=-2$ and $b=5$, so the x-coordinate of the vertex is $x=-\frac{5}{2(-2)}=\frac{5}{4}$.

Finding the Vertex of the Parabola

To find the y-coordinate of the vertex, we need to substitute the x-coordinate into the function. So, we have $f(\frac{5}{4})=-2(\frac{5}{4})^2+5(\frac{5}{4})-4$. Simplifying this expression, we get $f(\frac{5}{4})=-2(\frac{25}{16})+\frac{25}{4}-4$. This simplifies to $f(\frac{5}{4})=-\frac{25}{8}+\frac{25}{4}-4$. Further simplifying, we get $f(\frac{5}{4})=-\frac{25}{8}+\frac{50}{8}-\frac{32}{8}$. This simplifies to $f(\frac{5}{4})=-\frac{7}{8}$.

The Vertex of the Parabola

The vertex of the parabola is the point $(\frac{5}{4},-\frac{7}{8})$. This is the point at which the function changes direction. Since the leading coefficient is negative, the function opens downwards, and the vertex is the highest point on the graph.

The Axis of Symmetry

The axis of symmetry of a parabola is the vertical line that passes through the vertex. The equation of the axis of symmetry is $x=\frac{-b}{2a}$. In this case, the equation of the axis of symmetry is $x=\frac{5}{4}$.

The Roots of the Quadratic Function

The roots of a quadratic function are the points at which the function intersects the x-axis. To find the roots, we need to solve the equation $f(x)=0$. In this case, we have $-2x^2+5x-4=0$. We can solve this equation using the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. In this case, $a=-2$, $b=5$, and $c=-4$, so the roots are $x=\frac{-5\pm\sqrt{5^2-4(-2)(-4)}}{2(-2)}$. Simplifying this expression, we get $x=\frac{-5\pm\sqrt{25-32}}{-4}$. This simplifies to $x=\frac{-5\pm\sqrt{-7}}{-4}$.

The Complex Roots of the Quadratic Function

Since the discriminant is negative, the roots of the quadratic function are complex numbers. The roots are $x=\frac{-5\pm i\sqrt{7}}{-4}$.

The Domain and Range of the Quadratic Function

The domain of a quadratic function is the set of all possible input values. Since the function is defined for all real numbers, the domain is the set of all real numbers. The range of a quadratic function is the set of all possible output values. Since the function opens downwards, the range is the set of all real numbers less than or equal to the y-coordinate of the vertex.

Conclusion

In this article, we have explored the properties of the quadratic function $f(x)=-2x^2+5x-4$ and its leading coefficient. We have found the vertex of the parabola, the axis of symmetry, the roots of the function, and the domain and range of the function. We have also discussed the importance of the leading coefficient in determining the direction and the rate at which the function opens or closes.

Further Reading

For further reading on quadratic functions, we recommend the following resources:

• Wikipedia: Quadratic function
• Math Is Fun: Quadratic Functions
• Khan Academy: Quadratic Functions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  

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Introduction

In this article, we will answer some frequently asked questions about the quadratic function $f(x)=-2x^2+5x-4$. This function is a quadratic function with a leading coefficient of $-2$, which means that it opens downwards. We will explore the properties of this function and answer some common questions about it.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point at which the function changes direction. To find the vertex, we need to find the x-coordinate using the formula $x=-\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a=-2$ and $b=5$, so the x-coordinate of the vertex is $x=-\frac{5}{2(-2)}=\frac{5}{4}$. To find the y-coordinate, we need to substitute the x-coordinate into the function. So, we have $f(\frac{5}{4})=-2(\frac{5}{4})^2+5(\frac{5}{4})-4$. Simplifying this expression, we get $f(\frac{5}{4})=-\frac{25}{8}+\frac{25}{4}-4$. This simplifies to $f(\frac{5}{4})=-\frac{25}{8}+\frac{50}{8}-\frac{32}{8}$. This simplifies to $f(\frac{5}{4})=-\frac{7}{8}$.

Q: What is the axis of symmetry?

A: The axis of symmetry of a parabola is the vertical line that passes through the vertex. The equation of the axis of symmetry is $x=\frac{-b}{2a}$. In this case, the equation of the axis of symmetry is $x=\frac{5}{4}$.

Q: What are the roots of the quadratic function?

A: The roots of a quadratic function are the points at which the function intersects the x-axis. To find the roots, we need to solve the equation $f(x)=0$. In this case, we have $-2x^2+5x-4=0$. We can solve this equation using the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. In this case, $a=-2$, $b=5$, and $c=-4$, so the roots are $x=\frac{-5\pm\sqrt{5^2-4(-2)(-4)}}{2(-2)}$. Simplifying this expression, we get $x=\frac{-5\pm\sqrt{25-32}}{-4}$. This simplifies to $x=\frac{-5\pm\sqrt{-7}}{-4}$.

Q: What are the complex roots of the quadratic function?

A: Since the discriminant is negative, the roots of the quadratic function are complex numbers. The roots are $x=\frac{-5\pm i\sqrt{7}}{-4}$.

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

A: The domain of a quadratic function is the set of all possible input values. Since the function is defined for all real numbers, the domain is the set of all real numbers. The range of a quadratic function is the set of all possible output values. Since the function opens downwards, the range is the set of all real numbers less than or equal to the y-coordinate of the vertex.

Q: How do I graph the quadratic function?

A: To graph the quadratic function, we need to find the vertex of the parabola and the axis of symmetry. We can then use these points to draw the graph of the function. We can also use a graphing calculator or a computer program to graph the function.

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

A: To find the x-intercepts of the quadratic function, we need to solve the equation $f(x)=0$. We can use the quadratic formula to solve this equation.

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

A: To find the y-intercept of the quadratic function, we need to substitute $x=0$ into the function. So, we have $f(0)=-2(0)^2+5(0)-4$. Simplifying this expression, we get $f(0)=-4$.

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

A: To find the vertex of the parabola, we need to find the x-coordinate using the formula $x=-\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a=-2$ and $b=5$, so the x-coordinate of the vertex is $x=-\frac{5}{2(-2)}=\frac{5}{4}$. To find the y-coordinate, we need to substitute the x-coordinate into the function.

Conclusion

In this article, we have answered some frequently asked questions about the quadratic function $f(x)=-2x^2+5x-4$. We have explored the properties of this function and answered some common questions about it. We hope that this article has been helpful in understanding the quadratic function and its properties.

Further Reading

For further reading on quadratic functions, we recommend the following resources:

• Wikipedia: Quadratic function
• Math Is Fun: Quadratic Functions
• Khan Academy: Quadratic Functions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton